 In consumer behavior today we are going to study the case of the duality between utility function and indirect utility function. Utility is the power of a good to satisfy the human one and generally we say that the consumer is always having a very wishful thinking just to want increase in the utility. But how that utility can be increased and if that utility has to be maximized or increased we utilize various approaches to calculate that. Keeping in view the indirect utility function we utilize the maximization of the utility subject to a constraint and that constraint tells us that all the expenditure that is spent for the gain of that utility has to be equal to the budget or to the income of the consumer. So, utility is maximized through the price of the commodities and with the income that is in the hand of the consumer. In this sense utility is maximized indirectly through the two variables that the price of the commodities and the income of the consumer. So, when we having the second approach of the utility maximizing in that sense we say that the utility is maximized when the consumer is having more and more of all the commodities he want to purchase. Means utility is maximized through the attainment of more of good X or Y or X1 or the X2 means along with we are going to have any utility function. So, either we can have the direct utility function or either we can have the indirect utility function. Then what is the difference in the duality and we can explain that that the interpretation of the Lagrange multiplier that we have many times utilized to calculate the change in the utility with respect to the change in the income. We say that the increase in the income or increase in the money income will be having an incremental effect on the utility. It means whenever the consumer will be having more and more of its budget he has a possibility that he will be on higher indifference curve or higher utility function. So, the effect of a change in the price and this price is the any amount of the consumer that he want means if he is having indifference curve for pertaining to X1 and X2 that when he is having two commodities we can calculate this with respect to either X1 and also for the X2. In this position if we say that if this is related to the change in the utility with respect to the change in the price of X1 then we will say that it is utility from the X and change in utility change in that commodity X with respect to change in price of X and if it will be like utility of from both X and Y then it will be change in amount of Y with respect to change in price of Y. But here we are having this generalized version that either there is change in the utility or the amount of X pertaining to this then we can have this lambda and this aspect. And here when we have calculated this part that change in this total utility with respect to price is equal to lambda and that is equal to change in M by Pi. So, the budget constraint it must be satisfied either in this way or the other that this budget constraint has to be utilized to satisfy the assumption for the maximization of the utility. And now coming to this other and if we calculate this amounts and substitute in the other equation we come to that point that gives us the rise identity means here the amount of the X that has been decided with the change is equal to the change in commodity with respect to change in its price and multiply by its total price. So, in this way change in utility due to change in price it is equal to the lambda into that respective amount. And in this way we can say that if there is any increase in the price of that commodity keeping in view the law of demand function we can expect a reduction in the purchasing power of the consumers money income and according to that there will be decrease in again in the its commodity demanded. And as per the Stafford-Lemma when there will be decrease in the money income we will be having less demand for that commodity. So, the lambda is the marginal utility of the money income that we have shown here and the product of that lambda and that particular amount of X and that is the rate at which utility varies with money income. So, it is times the rate at which the purchasing power of the money income varies with PI and so in this way product yields the rate of change of utility with respect to PI. Here we have substituted both conditions of utility maximization through the direct function and at the same time with the indirect function and we can have the same assumption on both side to reach on the same point. And that same point it tells us that utility either via the maximization process or either via the price and the income it reads on the same point that indirect utility function it is also increasing in the money income and we if we invert the utility function that is when we say this utility is the function of prices and income to obtain the expenditure function and in this way the expenditure function it tells us that expenditure function M or in some examples we can utilize this as E also it is equal to again the prices and this U and here this U mean the utility or we can say that expenditure that will be required to at least ascertain that particular or the given amount of the utility or if we say that in the direct utility function the prices and the income will be the amount of utility and if we invert it then we will require a particular amount of expenditure which will be required which will help us to get the utility along with the given level of prices so keeping in view this situation we can reach at the same point via the expenditure function or direct utility maximization and in the same way both functions are dual to each other and they contain almost all same information that can be utilized to have all the assumptions for the utility maximization so when each of these T function that we have utilized either utility maximization or either the expenditure function or the indirect utility function they have the same information and they can be best utilized for the representation of the consumer representation so this tells that if we are having information related to the prices of the commodities or we have a consumer's income then we can derive its utility and if we can calculate its expenditure function along with its direct utility then we can calculate things. Thank you.