 here we are going to derive Marshallian demand function. And Marshallian demand function as we have already studied that this was the concept given by the Marshall and it is expresses the relationship between the quantity it demanded of a quantity and at the same time its relationship with its the price. So actually this provides us the solution that how with the change in the price quantity demanded of any commodity it will be changed. So when we see this it means we can calculate that quantity of the commodity that will give the optimal choice bundle to the consumer. And if we say in Tadalfa's case it is that if the consumer has two goods X1 and X2 and they are using their consumption bundle when their respective prices have prices 1 and 2 its income is constant. Whenever now there will be change in the price then its demand changes. Because of this change in demand its actual new optimal bundle will change. It means now the consumer will have the different utility level. From the first utility level its utility level or in difference curve level will change. If we are able to calculate the change in commodity demanded we can measure its quantity. So we can assess that its optimal choice bundle is either because of that it is better off or it is less. Now if we have to measure it then how we can do it. So whenever we have to decide that what is the actual amount or what quantity which gives the consumer optimal utility or maximizes its utility we have to measure its commodity so to measure its quantity we can take the help of utility maximization approach. Now utility maximizing approach if we explain then it means there is demand of commodity X which we already talked about depends upon various prices and income. When we have to utilize the utility maximizing approach then from this diagram we can see that the consumer is having this in difference curve that depicts the various bundles available to the consumer and these are bundles are on the same in difference curve mean consumer is indifferent among all these bundle but he select only this bundle A because it was tangent to his original budget line. Now suppose now that the price of the commodity X has decreased making possible for the consumer to shift to this new budget line or to pivot to the new budget line making him possible to have a new optimal bundle B and if the price of X decreases further again a new budget line is pivot out with the flatter slope and the consumer is having this third bundle T. So in this form we can see the consumer has shifted from bundle A to bundle B and then to bundle C in response to the change in the price and for these respective prices we can see that only the price of that commodity X has changed. When we measure the change in quantity demanded with respect to these prices then we can plot the relationship between that quantity demanded and this price. So when we plot this again we are having quantities demanded on X axis and the same respective points of the combination and their prices on Y axis and we join this then we can have this demand of the consumer and we can have the demand function. If I explain this in this form together then we can say that these are the factors or the points that have affected the consumer through these prices and now for these three the consumer have the similar points joining and consumer have expressed this in the form of this demand curve. But here this is the case when the consumer it was dealing with a normal good it means this relationship it can be different for the different nature of the commodities. So if I have to derive this demand curve for the normal good so I will utilize the same price consumption points and the same points derived downwards and explained on the vertical axis with the price. This demand curve will be explained for the normal good and it will show that the price consumption curve is moving upward and the demand curve is moving downward. If we will include the case of the inferior good in the inferior good again the price consumption curve it is upward but the backward bending and with that backward bending we can see now that the quantity demanded in response to the price fall has not increased rather reduced. So in this form we can have now that the consumption of the good has reduced and the price consumption curve is backward bending. When we say the neutral goods neutral goods we have already studied means these are that type of the good on which there is no change of the price change so whenever there will be any type of this good we will deal and the price of X if it falls budget line it pivots outward for this so it means only the quantity demand budget line has shift like this but the consumer has not shifted its demand from this point rather from this bundle to this bundle A to B consumer shift in a manner that the commodity demanded of X2 might be or the Y has changed but for the X it remains the same so it shows the price consumption curve that is the vertical form and we have the same type of the demand curve means that is vertical without any key. So Marshallian demand curve when it is derived mathematically we can have that we are having this utility function when we are having this multi good case that is our objective function and subject to our budget constraint that is our constraint so we can utilize the Lagrange multiplier model means where we can say this objective function plus lambda into subjective function and this Lagrange is now derived at the first order condition and through the first order condition now we are having n plus 1 equations and number of X and 1 for lambda so we are having this multiple level of the equations and change in utility due to change in X change in utility due to change in X2 and likewise we are having one equation equal to lambda and when we equate this we may have change in utility due to change in X1 equal to lambda into P1 and here we can derive like this so either we can have any type of equation where we can equate them with the lambda or we can equate with the price of 1 now coming to the point we have one equation where we can have this utility equation 1 divided to the modern utility equation with the other and it gives us this price ratio and from the same we can have the value of lambda and from this equation we can come up with that this price when it is placed here and lambda up to this we can have this that lambda is actually that value that determines that what will be the change in utility with respect to one unit of price and likewise price will be the change in utility for one unit change for one unit of lambda so that it gives the economic valuation of the commodity and from here we can check that how much consumer is willing to offer or to pay for one additional unit of the commodity so when we substitute this value of the price into our budget constraint we come up with the amount of X1, X2, X3 so these like range help to determine the Marshallian demand function