In this section, we are going to understand the relationship between

the Beta parameters that we have previously defined

and the concept of elasticity of substitution.

In this case, in the context of a composite indicator.

Please remember, in the previous slides,

the formula of a composite indicator as a weighted average of

different partial indicators weighted and as a function of these Beta parameters.

In this case, what we are going to understand is how, when,

and how these Beta parameter is related to

the elasticity of substitution between the different partial indicators.

Let me be a bit more precise with what I mean by elasticity of substitution.

The basic definition we are going to give here is that the elasticity of

substitution between the partial indicator J and the partial indicator J prime is

just the amount of partial indicator J that I have

to give up just to get an additional unit of the partial indicator J prime.

The mathematical definition of this elasticity of substitution

is given in this slide and where I define it as Delta_12.

So then if I have two partial indicators one and two,

the elasticity of substitution between one and two is the derivative of

the log ratio of both partial indicators with respect to the log of the margin of their,

a marginal rate of substitution.

And in technical, in other fields within economics,

elasticity of substitution has a meaning or has

a sense in terms of ratio of prices variation.

In this case, we are not going to use it,

but it's also interesting for you to know that it exists.

The reason of these of the appearance of the of

these ratio of prices is that as you probably know,

first order conditional for the maximization of

a standard utility problems give some equilibrium condition

which is that the marginal rate of substitution between

two goods is just equal to the ratio of their prices.

So in this expression, Delta_12,

you can substitute the marginal rate of constitution T1 to one by the ratio

between P2 and P1 and you will

obtain the standard expression of the elasticity of substitution.

Just now going to the next slide,

consider just a very simple example.

Define the composite index,

only for two partial indicators,

a slide is define and then it is very easy,

you can do it as an exercised because it's rather simple to do it.

You can get the following transformation.

If you just take the derivatives and you work in them,

in using very simple mathematics,

you will come to the expression Delta_12 is equal to one minus one minus Beta.

So as you can see here,

this is the relationship that we have between

the elasticity of substitution and the parameter a bit.

If you come now to the next slide,

you will find a very nice interpretation in terms of graphics.

You will see that basically this elasticity of substitution is based or is

just the shift over the constant core of the composite index.

Let me just repeat the previous definition and

give just to settle concept in a proper way.

As you can see in the graphic,

you have two points: A and B,

in this one point is the starting point when you have

a ratio of quantities x1 and x2 and a marginal rate of

substitution between x1 an x2 or if you want a ratio of prices.

Now, if you go on the curve,

you will come to position B.

Of course from going to A to B,

what you have to do is what?

You have to give up some parts or give up an amount of one of the partial indicators.

In this case of indicator x1 to give one additional unit,

one extra unit on the indicator x2.

But please, keep in mind,

that this is given up from one side to get from the other side

is done keeping constant that composite index value.

This is the crucial issue,

is just the shift along the curve,

the curve or the composite indicator does not experienced any shift.

Okay, going now to the next slide,

we can just see

how the form of the composite index changes according to the value of Beta.

So, for example, when Beta takes values equal to one,

then Delta takes value equal to infinity and we get the standard weighted average.

The standard weighted average of the composite index.

If Beta takes value equal to zero,

then Delta will be equal to one.

As you can see,

we get a different type of composite indicators.

Of course for Beta smaller than one,

then Delta will be greater than zero then we get a concave function.

And finally, it's also

a very interesting version when you take Beta equal to minus infinity,

then Delta will be equal to zero and then it won't be possible

any substitution between the different partial indicators in the composite index.

Okay going now to the next slide,

we will see that or we will show you some examples for different composite indexes.

For example, for Beta equal to one,

we have the Life Condition Index,

the Commitment to Development Index,

and other indexes as you can see this slide.

For Beta equal to minus one,

you have the Gender Development Index.

And for Beta equal to three, you have the Human Poverty Index.