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So as we saw earlier, the first step is to see what other dimensions we want to

include in this index.

The human development index has three dimensions.

One is, the first one is health, the second one is education, and

the third one is decent life, which is going to be proxyed by income,

but I'll come to that later.

So we have three dimensions, health, education, and living conditions.

So the next step is what are the indicators that describe well-being or

that reflect well-being in these three dimensions?

As far as health, they have chosen one indicator, the UNDP.

And in fact it's an office which is called a Human Development Report Office,

HDRO, that publishes these indices, that publishes these index annually for

all countries of the world for which there is data available.

And in fact, every year they publish,

not only this index, but also other indices for different countries.

And they kind of, the report contains information

on the general information on how the world is doing in terms of well-being.

The different countries or the different regions of the world,

how are they varying in terms of well-being?

But also, every report concentrates or gives particular

attention to a particular theme, like for example, gender equality or

environmental sustainability or work well-being and so on.

So every year there is a particular focus given to a certain theme.

So let's come back to the index.

So it has three dimensions, health, education and living conditions.

Health, there is one single indicator which is the life expectancy at birth.

For education, we have two indicators, mean years of schooling and

expected years of schooling.

For the living decent life or living conditions dimension,

the per capita gross national income is take as relevant indicator.

So let's start with health.

Health life expectancy at birth, what is life expectancy at birth?

I'll just simply put, it's just simply the average number

of years that a child born in country is expected to live.

So the average number of years of life of a child at birth.

That's the life expectancy, so the unit of measurement is years.

Let's say, for example, 80 years, or 65 years, or 95 years.

So it's an indicator which is measured in years.

Then we go to education, there are two indicators for this dimension,

mean years of schooling and expected years of schooling.

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Here maybe it may be important to note that

when you have different number of indicators per dimension,

it is all the more important to have this multi-level or two level reasoning.

As to what are the dimensions which we want to include, and

what are the indicators.

If you don't have this two level reasoning,

you run the risk of giving undue importance to one dimension, for

which you may have more indicators than others.

So it's better to allocate the weights of cross dimensions first, and then to look

at indicators within a dimension and distribute the weights within a dimension.

So in education we have two indicators, mean years of schooling.

Again mean years of schooling is simply the average years of education of

the adult population in a country.

And the expected years of schooling is the average years of education that

a child enrolled in school is expected to have at the completion of his or

her education based on the current enrollment rate.

So based on the current enrollment rate, how many years of education

will a child have after completion of education, on average.

So that's again these two indicators are in years.

One can imagine that the first one, that the mean years of schooling is not, or

education, is not going to be moving very fast because normally,

the adult population in the country has already completed its education.

And so that's not going to be changing unless the generations get renewed.

So that is why they also added the other indicator, which is the expected years of

schooling, which will move faster because at more and more children

get enrolled in school that is a likely to move faster than the first one.

That's also these two indicators are also in years.

And then we have the third indicator dimension for

which there's one single indicator, which is the gross national income per capita.

And that is measured in dollars, US dollars and in PPP US dollars, that is

purchasing power parity US dollars, which is basically a kind of currency,

I mean conversion of all currencies into comparable US dollars.

That is in such a way that we can compare

the per capita income of different countries in this PPP US dollars.

So now we have for three dimensions and four indicators.

We need to combine them, for the two dimensions for

which there is only one indicator, there's nothing to be done.

But we notice that we have three indicators in terms of

years as unit of measurement and the last indicator which is the per

capita national income in terms of PPP US dollars.

So how do we combine something which is in years with something which is dollars?

So first ting we need to do is to convert them into a same

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unit of measurement so that we can combine them at a later stage.

So how do we do that?

This process is called a normalization process.

And the normalization that is chosen or

that is used in the construction of the HDI is as follows.

So you take for, so the index, the normalized index if you want is given by

the value of the indicator minus a minimum value divided by maximum minus minimum.

So what are these maximum and minimum values, or where are they coming from?

So for each indicator, we need to set minimum and a maximum.

So what is this minimum?

The minimum is that level of achievement at which or

below which the country is supposed to be at zero level of achievement.

And the maximum is that level of achievement at which, or

above which, the country is supposed to be at a maximum,

having attained the maximum level of achievement.

So let's take life expectancy for example.

The minimum and maximum have been set at 20 and 85 respectively.

So if a country is at level of 20 or

below 20, then it hasn't had any achievement in this dimension, so

the achievement will be zero.

And if for a countries at 85 or above 85, it considered to have achieved

the maximum level of achievement for this indicator, so that'll be one.

So this normalization process basically what is does is to project

the value onto a scale, a zero one scale.

So maximum achievement being set at one, and

the minimum achievement set at zero for the normalized value.

For the mean years of schooling, this maximum is set.

There are natural minimum and maximum possible for these indicators.

For example, years of schooling, obviously the minimum is zero and

the maximum is set at 15 for the mean years of schooling.

Which is the years of schooling, average years of schooling of

the adult population as I said, let's say of the older generation.

And the expected years of schooling has a minimum also at zero but

a maximum of 18 because is the years of schooling

supposed to reflect the years of schooling of a younger population.

Who are normally, if you go in overtime, we expect that

the younger generations have higher levels of education than the older generations.

And the last indicator, per capita income, the minimum is set at $100 PPP,

and the maximum at $75,000 PPP dollars.

These are the major steps involved in the construction of a multi dimensional index.

Now let's see how this normalization works and

what is the meaning of this normalization.

So we set for each indicator, we set a minimum and a maximum.

So let's say the minimum is here and the maximum is here.

So remember that this is a level of achievement at which or

below which the country is supposed to be at a zero level of achievement,

zero level for the index normalized value.

And the maximum value is that level at which are above it,

the country is at a level of one maximum achievement.

Now, so this is the maximum possible achievement for any country.

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at what is the percentage of achievement that the country is at

compared to the maximum possible achievement that it could have.

So if you take this A, So

the A is basically value minus the minimum value.

The value for that country minus the minimum is the length A, and

if you look at the length B, it's simply the maximum minus the minimum.

So where is this country compared to the maximum level of achievement that it

can have?

So that is the normalized value of every indicator.

And this normalized value as you can see goes from zero to one

because anything at the minimum or below the minimum is zero.

Anything at he maximum or above the maximum is one, and

any other value is between zero and one.

So we are projecting this value onto a scale which goes from zero to one.

So that we do it for each indicator.

So the life expectancy is converted into an index, a normalized index in this way.

The education dimension has two indicators we set, the mean years of schooling and

the expected years of schooling.

And each indicator has its own maximum and minimum.

So we'll do the same operation for each of these indicators, and so

we get two normalized indices for the education dimension, and

the overall education index is taken to be a simple average and

arithmetic mean of these two indices.

The income index is also normalized,

the per capita national income is also normalized in the same way,

but using the log of the value rather than the value itself.

And I'll come to that in slightly

in a short while to say why we use the log value.

So we have now after this normalization procedure we have one index for

health, one index for education which is in already an arithmetic mean of

the two sub-indices if you want.

And then one index for income.

And the human development index is simply constructed as

the geometric mean of these three indices.

So the formula is that we do the product of these three indices and

raise it to the power one-third, that's called a geometric mean.

So we saw different types of means are being used in this process,

one is an arithmetic mean and one is a geometric mean.

And that has to do with the degree of substitutability that we want to have

among the different elements that make these means.

So an arithmetic mean normally allows for perfect substitutability, and a geometric

mean for much less substitutability among the different dimensions.

So the overall HDI is the geometry coverage of the three dimensional indices.

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