 module 94, chapter is consumer behavior and we are going to study expenditure function. Up till now many times we have discussed this topic, the expenditure function or the minimum expenditure. So now we have to tell it in detail that what we mean by the expenditure function. Actually when we are dealing with the maximization of the utility of a consumer that utility on one side is having that concept that the expenditure or the income of the consumer it is kept constant and then we are going to derive through our first derivative or the optimization techniques or even the practically that how the consumer will be able to attain the maximum level of the utility with that given level of the income budget or with that given level of the expenditure. But on contrary that can be solved by the problem in the dual form that if that level of the utility that we have picked or that the consumer has wished for if that is kept constant that how much amount of the expenditure will be required and that expenditure should be the minimum expenditure that will make possible to the consumer to have that same level of the utility. If I say this in simple words then we will say that there are two elements of one share. On one side we say that the consumer wants to get his utility more and more. He wants to get the amount of the utility that he has or what we call his demand he wants to get more and more of his utility but for that he is always limited. His demand is always limited. That given, that expenditure, that expenditure, that expenditure is his expenditure and in that expenditure he maximizes his utility. And on the other side if we look at it then we can say that the utility that he has given he has decided, he has decided. We call the utility that he has given so much utility. That means if he says that I have made a budget by looking at the expenditure of my house or he says that my needs are so much that in a month I have to take so much flour, so much butter, so much ghee for my family. My electricity bill has to be so much and this is my total utility. So for this, I will require so little expenditure or budget. And because he also attaches with prices and similarly when we go to utility even there, he attaches with Ashiya's and Amdan's prices. So if we look at the problems of duality or duality, then we can go to maximization of utility or instead of maximization of utility we can go to minimization of the expenditures function. So we will be working on one thing. So when we talk about expenditure function, then we will say that these will be at least, at least, a consumer would like to fulfill their particular utility. So that will be equal to the minimum expenditure. And we say that that will be equal if we look at it here, we have shown in the equation for the given amounts of the prices which are prevailing in the market at that time. But depending upon that utility, that is fixed for that consumer. So this definition it shows that expenditure function is particularly and its indirect utility function of the consumer, they are the inverse function of the each other. So either we have to work on maximization of indirect utility or either we have to work on the minimization of the expenditure. So now coming to the next point if we say that there are the two aspects on one side we say utility maximization, so if we go to the other side then we say that minimum of the expenditure. So whether we achieve one thing and whether we work on the other thing, we will reach the same result by working on both things. Now if we look at it that when we are working on utility maximization, then why do we need to work on expenditure function? So if you look at the demand in this, then many times expenditure is in front of us in a very clear form. Because income or budget is the thing that we have in the average. And instead of the prices, when we go to the market, then the thing that is clear to us is that with the drop in the price of the prices, we see how much expenditure I can make in my average expenditure. So from here if we look, then that profit is easy to adjust according to its utility and according to its profit. The demand of the asset, it is easy to adjust through the expenditure that is being indicated. So this comes in front of us in a second form which gives us an easy way that if we maintain its utility level, and if we have to assess the minimum expenditure, then since the minimum expenditure is keeping direct balance with the prices, the assessment is being done easily, and it is easy to calculate any average cost. So we can use the minimum expenditure function more and this can be helpful in making our average cost with ease. Thank you.