In this section of course and the next one, will be devoted to a study, probably the most important feature within the context of component's indicator, which are the weights. The weight functions are definitely the most important choice to do for the researcher when you want to elaborate a composite index. Let me just in the next two slides, give you a couple of examples of what is the sensitivity of the values that take the composite index to the choice of the weights. Let me also show you in this part of the course, how can we understand the meaning of the weights in terms of marginal rates of substitution between the different parcel indicators. And then, for the next part of the course, for the next section, we will discuss different types of weights, how to choose them. Okay. So now, let me just, if you go to the next slide, you will show a graphic. This graphic is taken from a very interesting work by Andrew Sharpe and Brendon Andrews in 2012. The title of the work is 'An Assessment of Weighting Methodologies for Composite Indicators: The case of the Index of Economic Well-being.' It is published within the context of the Center for the Study of Living conditions. If you see this graphic, you can see a very interesting feature for the index of economic well-being. Just take the baseline graphic which is the evolution of this composite index for Norway. When we take, as you probably remember, the composite in this index for economic well-being have different dimensions and it always rest of course the problem of choosing the weights for these dimensions. If we take equal weights, that these 0.25 for consumption, 0.25 for wealth, 0.25 for equality, 0.25 for security, you can see the blue line, the base line. However, if we change the weights in as in alternative one, two or three, you can see that the composite indicator changes in a rather important way. The question will be always the same, which are the right weights? Which are the weights that better approximate the phenomena we want to explain with the composite indices? Let us try to shed some light to these relevant questions. Also, if you go to the next slide, you will see a very important issue. According to the weights we are using this index of economic well-being, you will see that the order in the ranking of the countries is different. For they equal weights, you see that Norway is top then we have Denmark, Germany, Belgium, but if we choose alternative one, even though some countries would remain in the highest places as for example Norway, there is a change in the ordering of others. So again, as you can understand, they issue of which weight do I choose is a very relevant question. Okay. Let us now go to the third slide and let us introduce some mathematical notation. Here, what we want you to understand is, what is the relationship between or better than saying that relationship, which is the impact of the choice of differing weights into the marginal rate of substitution between partial indicators. As you probably remember from the previous section of the course, the marginal rate of substitution between two partial indicators, J and J prime, is basically the amount of J prime of quantity of J prime, I have to give up just to get an additional unit of J. Basically it has substitute ability relationship. Let me just show you the following in the slide. The marginal rate of substitution between these two partial indicators, J and J prime coming within just as a ratio of the derivative of the composite indicator with respect to these two partial indicators. However, if we assume that the composite index has a function or are a standard expression as the one we defined also previously, then it's easy to show that the marginal rate of substitution can be written as a product of these three factors you can find in the slides that are colored in differing ways. In this three factors, you have first, as you can see, a ratio of weight between the, exactly or more precisely the weight of the J factor and the weight at the prime factor partial indicator. Then you have a second term which is the derivative of this partial indicator with respect to this quantity. And then you have a third expression that is taken to the power of 1- Beta and which is just the ratio of the two partial or transformed partial indicators. What is the relevance of this three term expression in what we are concerned with? Let us go to the next slide, and what we are going to do now is to analyze any of these terms by separate. The first term as I told you before is a ratio between the weight for dimension J, and the weight for dimension J prime. So, if we increase the weight for dimension J, what are we doing? What we are doing basically is we are increasing the marginal rate of substitution. So basically what we are saying is, "Okay, we need to increase the number of units of the dimension of the partial indicator J prime, that we have to leave in order to get an additional unit of the partial indicator, J. "That's exactly the impact of the weights on the marginal rate of substitution. So, as you can probably understand, the way we choose the weights, it's affecting really the substitutability or in between the different partial indicators that we are having in the formula in the expression of the final composite index. Then if we go to the next slide, you can realize also that we have indeed there the second term in these three terms formula. In this second term, you can see the following, the steepest is the transformation of the partial indicator in J. The largest is also the marginal rate of substitution between J and J prime. So of course, the steepest transformation is for this partial indicator is also affecting definitely the marginal rate of substitution. And finally, as you can see there, the inverse transformation of the ratio of both partial indices is also relevant, and is relevant also through the value of Beta. Let us go to the next slide and let us make a very simple exercise. Let us ask the following, under which conditions the expressions that this three formula expression becomes just a one formula expression? Let me explain myself, under which conditions the marginal rate of substitution between J and J prime is just equal to the ratio of weights. In order to do get, or to do so, what we need, as you probably realized just looking at the formula, if we make beta equal to 1, and we make that the derivative of the transformation are equals, and we establish these two conditions, then we will get the final expression we are willing to, that is the marginal rate of substitution between J and J prime is equal to that ratio of these weights. As you can see then, this two restrictions we have imposed to get this final expression, and repeat them, I repeat them. First of all, beta equal to one. Second, the derivative with respect to the transform values of J and J prime are equal. If these two conditions are fulfilled then the choice we've made for the weight is really crucial to in terms of substitutability between partial indicators. The problem basically is that these two conditions are usually fulfilled by many composite indices. So basically what we are saying is that, weights do play a crucial issue when we are talking about substitutability in composite indices.