 Module 99, under the chapter of consumer behavior the topic is expenditure function properties and this property is the expenditure function is increasing in utility and this utility is particularly as we have shown that it is with respect to the given expenditure. So expenditure function if we see that whenever there will be any type of the commodities expenditure it has to be particularly attached with certain commodities quantities. So whenever there will be one commodity or the two commodity or like these three and the consumer has to select a bundle of the commodities there will be whatever the level of the prices there will be a particular amount of the expenditure and if more of the utility has to be attained for the given prices then for those given prices if more commodities has to be purchased then more expenditure will be required and it is the same as if you are going to buy something in the market, you have paid the price and you are buying anything you buy a dozen but if suddenly you want that no guests will be at home and I have to buy two dozen instead of a dozen, then the price you have decided with that now your utility level is increasing, your demand is increasing, so you will have to spend more money. So whenever for given prices that is the main point here we are not changing the prices. So for those given prices whenever more of any commodity has to be demanded we have to spend more. And if we use the example of Lagrange Multiplier which we have used for the maximization of utility and minimization of expenditure and if we use it then if we say that where our utility is positive we use it but our utility is not zero, if we go to our minimization of expenditure then we see that our Lagrange Multiplier it asserts us that that is equal to the change in the money expenditure with the related to the change in the utility. So whenever we change our utility then with respect to our amount of money or expenditure will also change and in the same way if we see then in our expenditure whatever is one share or two or three or more it can be more than that, everything is under the market price expenditure. So on top of that what we are looking for is its related cost, for one additional unit if we decide then we will say that every additional unit taking any of the x, y, z which is on top of that will be its marginal cost and this marginal cost when we see this, then this marginal cost tells us that the rate of change of the minimum expenditure because whenever we increase one thing then its minimum expenditure or its marginal cost will be added up. So that is why those different marginal costs added up because the total cost of expenditure will be for the purchase of various consumer items that will increase for the given level of utility or for with the passage of increase of the utility. So if we see the rate of change of the minimum expenditure which we have shown here with respect to required utility level we say it can increase as we enhance the utility level. And if we see from this, then from the small mu if we show it, then this is reciprocal which we use in consumer function in Lagrange multiplier, an amount notation that is called lambda. Because the lambda here is cross-poned for utility maximization and when we used to talk about that utility maximization then we say that lambda shows us any commodity per unit price. If we say that if what is lambda then we say that lambda is equal to the change in utility with respect to the change in price. So if we see that this means that under a unit price how the utility of any commodity changes. So if we reverse it and see its reciprocal then we can say that basically if we reverse it then we see that our price goes up and if we convert that price or give other words of our cost or expenditure then we say that now it is change in the minimum expenditure with respect to utility and now we give the notation of that small mu so it means it is the same. So now it will become here become marginal utility of income, we have here marginal utility of price and it will come to us that with the change in utility how are we doing the marginal aspects of our income and if we see in this then the theory of ordinary utility on which we are taking things under the order means we were ranking them under the indifference curves we were looking at it so it allows us that the sign of mu is established because in ranking whenever we take its terminal point or we take its tangent point then we have Asia's respective prices or commodity quantity both are present and the product of these two can always take us to its related expenditure. So we cannot say that mu is either necessarily increasing or decreasing with utility but it will be related to that aspect that will be shown by that particular utility or the ordinary utility function. So it can be that whenever we say that we have to increase utility for given prices because if prices are not changing then we have to go to it that yes the related expenditure function it will increase.