 In the consumer behavior today we are going to study the consumer choices and in consumer choices we were already studying that what are the various properties of the expenditure function. This topic is pertaining to the expenditure function having the property that it is increasing when utility is increasing. As we have already studied various parts of our previous lecture that whenever the consumer wants to increase its utility or want to have more there are only two aspects that either he can have more budget or either he has to substitute certain commodities with one another but within the same budget line. But here if total utility has to be increased this is only and only possible that the consumer is having higher budget level or in other way we can say that a minimal budget is compulsory to have achievement of a particular level of the utility. When we were utilizing the Lagrange multiplier then we derived the lambda and that lambda told us that lambda is equal to change in utility with the change in the price or it was just equal to the other way we can say that it was the marginal utility with reference to the income. But if we inverse that lambda then we come to know that that is just having one relationship in which the change in income with respect to the change in utility is there and that is called the marginal cost of the income. It means how much income will be changed just to have our desired change in the utility. When consumer wants to have increased utility then there should be a positive increase in its income at the same time to at least equate that change in the utility. So this property of the expenditure function is mostly fulfilled even through the indirect utility function or either we are utilizing the utility function in both ways. Then if we have to utilize the utility maximization equation we have to exert that rule that the utility will be maximized keeping in view certain subjective budget function. And if we utilize the cost minimization even then we have to maximize the utility and in that way when we utilize the cost minimization we will utilize that the cost is minimized keeping in view a particular level of the utility. So in both ways we can have that for the higher level of the indifference curve a consumer requires more budget to achieve that higher level of the indifference curve or the higher utility level. Mostly the assumptions underlying the ordinal utility function theory it allows that the sign of the U that shows the marginal cost of the money that it shows that we cannot say that this mu is necessarily increasing or it is decreasing because both are possible for the different permissible utility functions that might be consumer want to decrease the utility or might be consumer want to increase the utility keeping in view the certain conditions. So the expenditure function will be increasing in the prices as well as we have already studied this and at the same time it will be increasing in the form of the utility. So if we has to increase our target utility or the consumer want to increase its utility then the constraint becomes harder to satisfy and therefore the cost of attaining that target has to be more and more. And at the same time minimum expenditure it requires to reach at the given indifference curve for that given prices and that given prices are utilized as a set risk peribus and is unaffected by any number. So when we multiply all these by the same number then the respective ratio will be given at the same level. This implies that a particular relationship between expenditure that is from the income M and the utility it remains the same but the properties we set out above they hold that for all the utility functions but only general restrictions we can place on the relation between M and U when there can be certain price factor is that it is mostly monotonically increasing in the way that if income will be increasing then we can expect the higher utility level and likewise if higher utility has to be increased higher income is desired at the same time. Thank you.