 Under the chapter of the consumer behavior, we are going to study the various property that an expenditure function has to agree with it. When we say the expenditure function, so definitely it will be related to the two things mainly, mean the commodities and its quantity. And those quantities have to be purchased from market at various prices. So, definitely there will be the utilization of the price. In our expenditure function, if we have to deal with the two commodities X and in this example, if we say that there are the two commodities, one we can say X prime and the other is X double prime. So, then their respective prices will be P prime and the P double prime. Now, these two prices that the consumer is facing. So, if we see that now the consumer has to shift or switch or not switch, we can say it has to substitute. Remember, I say that the consumer, between the two commodities, by changing their price ratio, is switching between them, that while moving from top to bottom, I am adjusting between these two situations that my minimum expenditure gives me maximum utility. But it is above the utility function. Now, if we look at the two commodities that we have mentioned, their amounts X prime and X double prime. So, one individual of these two commodities, their minimum expenditure function will be that how much expenditure function we are spending on X prime and how much we are spending on X double prime. And if we look here, we see that after two prices P prime and P double prime, we are taking a scalar quantity and scalar in the sense that if we say that is the amount K and it moves from zero to one, means it is not a negative, a non-negative scalar. And with this K, we are going to attach with these two commodities and the prices in the sense that one price P prime, we say that if that parameter, means K P plus added, we are doing 1 minus K into P double prime. And if we sum up these two, then sum up these two and we will get the price P bar. So, we have to adjust on that P bar on which they stay somewhere between these two points with the minimum expenditure, it attains its maximum utility. And if we go to this, then we say that the property of concavity tells us that if we see that the more prices increase, then expenditure increases. So, if we have to minimize the expenditure, then we will have to see that the minimum point like this, we will have to stay inverted from there. So, minimum expenditure or the minimum amount of the money that is required for that minimum expenditure function of P bar when utility is kept constant, it should be either it should be equal or at least it should be greater. And what will be greater is on both the points which we have utilized for P single prime and P double prime. And if we look at this, then we see that both the points are there. On both the points, if we say that one amount we have X and the other X double prime. And on both the points, when we are utilizing our price, we are taking P prime and P double prime. So, the expenditure function requires us that on one point, the expenditure function of P single prime will be this one. And the minimum expenditure at least on P double prime will be this one. And now between these two, when we want to see, what we have to do now, we have to do P bar prime and for us, we have to see at what level this one is coming to us. And now when we go for the next point, we see that if there is a price vector B, because if we have to see this price vector B and we have to go to that point which we are showing in our next diagram, then we have to derive this and how it will be possible for the consumer to switch from one part to here. And between that, its movement along the same utility line. As we said in the previous slide, that when we look at the concept of respective expenditure minimization, then the consumer already who is attaining with his specific points, if we look at that, then we have a condition for that. If we go here, then this is its first P prime price, but X bar is the commodity or the amount of commodity which will be taken towards the minimum expenditure. So, if it is like this, then if we decide here, then this is its initial point, but if it has to come to the point of X bar, then if we go to this, then it should be greater than the previous. And in the same way, if we talk about this, then we say that now if X bar commodity is added to this, it should be either equal or greater than this point. Now, while looking at these two points, let's look at the next one. So, we have solved the scalar points K and 1 minus K. We multiply these two scalars with these two inequalities. And when we multiply with these inequalities, then we see as per rules, we multiply this K and multiply the other left side with 1 minus K. After doing this, we have taken its constant factor X bar here and we have summed up that equation. Likewise, the right side of inequality, we have also multiplied it in the same way and we have come up with this situation. So, now if we multiply these inequalities with the mayor and the sum up, then the equation we see here is something like this. Now, if we look at this in the form of a diagram, then we have talked previously that if we are at the point of X prime, then definitely its price was P single prime. And if we are at the point of X double prime, then its price was P double prime. And now, if the consumer has to adjust its utility in between of all these where he can have the maximum utility, with the minimum expenditure function and if he goes through the concavity, then look at that, he is moving from this point to this point and look at that from this point, then it will come to this point. And for this commodity X bar, and for this we have decided its price, P prime. So, for this P prime, P bar and X bar, which is the formula for minimum expenditure, it shows that moving between these two inequalities here, it will reach in this form. Otherwise, if we look at its original P prime and X bar, then its budget function should be here. And similarly, if we say that it is not moving here, rather, with the increase of P price, the expenditure function if it increases, then instead of this point, if we look at this point, then its higher level of budget comes. So, with the increase of prices, expenditure function always, in that concavity means higher print shows. This is its property, which we have decided here. But when we go to its minimum expenditure, then if we look at it, then it shows us that that is though higher, but being the minimum expenditure, we attain a higher level with minimum expenditure. Thank you.