 Today's topic is Marshallian demand function and what are the properties of the Marshallian demand function? Marshallian demand mostly tells us the relationship between the quantity demanded and its price and it tells that there is always a negative relationship between the price and the quantity demanded and here the consumer derives his utility through the consumption or not directly through the consumption rather through the expenditure that he incurs on the consumption. So he conceals the other fact and he prefers to take the utility through the indirect form. Usually when we say that there is a property it means that these things should be the particular part of that either parameters or the policies. So when we say the properties it means that they are the very necessary things. So whatever we are going to conclude related to the Marshallian demand function if any of this property will be violated that general conclusion will not be followed. The first property that we are going to discuss it is the adding up. What we mean by the adding up? Generally the property of monotonicity it tells us that the consumer is having a desire to grab more and more of any commodity and he is desirous to get more of the quantity that he has already preferred. But when we add up all the things and when we add up all his desire but there should be a limit on this and what will be the limit? So that limit tells us that whatever the desire of the commodity to a consumer is there when we add up all these things it should at least be equal to the budget line or it should be less than the budget line it should not be more than that. So when we include the properties and we take the multi good case and we say that the consumer is having the demand function for commodity X1 or the commodity X2 or the commodity X3 up to commodity XN and then we multiply that commodity with their respective prices it means now the price of the commodity X1 multiplied by the X1 it will having the expenditure that the consumer is incurring on the purchase of that commodity. So if we say that if the consumer is buying all the properties of one, two, three, so when he gives the value of every share to him then when we multiply the value of his commodity with quantity then he comes in the form of a complete expenditure on that share and when we add up all the assets in his consumption bundle then he comes in the form of a total expenditure so it means that will be the total expenditure so here the expenditure incurred by the consumer for the purchase of all his desired commodities it should be within the budget limit and it should not exceed the budget limit. Second property that we are going to deal is the property of homogeneity and what we mean by the property of homogeneity is that if we are going to increase the units of the demand function means the prices price of good X1 price of good X2 and any other and if we are going to make them double by all the total function is multiplied by the same unit so if we are going to double the prices and at the same time if we are going to double the income then the output level of the consumption that the consumer is going to demand will be the same as like the previous so the demand function if multiplied by any unit t it will yield the consumer at the same demand level because income and the prices they both will be increased or decreased by the same unit so whatever will be the change that will be offset by the incremental part of the expenditure through the incremental part of the income and now coming to the example of the Marshallian demand function if we take the function of a utility where the two commodities X and Y they are utilized and now they are in the form that X have the elasticity of 0.3 and Y is having the elasticity of 0.7 and that this demand function if we are going to derive through the quantities demanded of X and quantity demanded of Y through any Lagrange function then we come up up to this point and now here we see that whatever is the shape of the consumption function or the utility if we just multiply the whole if I am going to put here this demand function and rather this demand function but if I am going to multiply the respective subject constraint that we are having here in the form that income is equal to price of X into X plus price of Y into Y so if I am going to multiply this by 2 or if I am going to multiply by this 3 whatever will be the form the resultant demanded quantities they will remain the same and the same relationship or the same behavior it can be exhibited by any other form of the demand Marshallian demand function that can have any form of the CES or the Cobb Douglas so even in this next equation we see that now the utility function or the demand function is something different but their budget line or the subject or the constraint is again this price of X into commodity and price of Y into Y and this budget if multiplied by the double or if multiplied by 2.5 or if multiplied by the 3 whatever but because this change will be having the same level of the change on income and the same level of the change on the their respective prices so the resultant quantity demand is will remain the same now coming to the third property that is called the negativity what we mean by the negativity is that if there is any budget constraint that enables the consumer to purchase a quantity of a bundle that is X1 and at the same time consumer is having the bundle of X1 and X2 or the Q1 or the Q2 but that budget makes him possible to purchase the commodities quantity X1 it means now there will be no other bundle like this or no other budget constraint that will enable him to purchase the quantity of the X2 if we look at it then we talk directly about consumption on its expenditure that if any budget which we want to take and that budget gives our consumer this affordability that out of the bundle of X1 and X2 is purchasing the quantity of X1 so we should keep in mind that along with it its affordability and the limit of the budget is also that apart from that it has no other purchase it is not possible so if we look at it then the concept of this negativity and the fourth the main thing is that is the slutski equation we have already studied related to this the slutski identity and here through the marshalia demand function now we can see that it makes possible to the economist to derive the slutski equation as we are having a single objective that we have to maximize the utility and if we have to maximize the utility at the same time we have to check that what is the change in the quantity demanded by the change in any price so when we measure this thing we utilize the two ways one is the utility maximization and at the same time we can utilize the other form that is the cost minimization and this part is the Hixian and this curve that we are drawing on the right side that is we are having through the marshalian demand curve and this marshalian demand curve that was having the dx into price of x and price of y and income and here now this income was substituted by the expenditure function so this equality of Hixian and the marshalian demand curve makes possible for the economist to draw that what is the amount of the expenditure required to attain the same level of the utility so when we take this two equal this part of the marshalian demand function this is further decomposed in these two parts and this says that this change it is in the two parts change due to price of x and this is the change that is due to the change in the expenditure so marshalian demand function through its decomposition provides the facility to the economist to have the analysis of the change in quantity demanded through the price via substitution effect and via income effect so we see that in this slutsky equation this right hand side that we have noted through our marshalian demand curve it shows that the changes in the price of x that we were dealing with they affect the demand of the commodity not directly rather indirectly via changes in the expenditure function so here the change in commodity demanded will be due to change of his expenditure that the consumer has to make in response to price change means if price has increased he has to adjust for that by increasing the expenditure or if price has decreased then how he has to adjust so that change in expenditure and then what is the change in expenditure related to the price of x and mostly this part is just equal to the commodity the previous that he has purchased so this makes possible to the consumer to have this income effect through the marshalian demand function and this mathematical representation of the income effect we are having here so this marshalian demand function makes possible to give the decision or to give the reply of our question that what will be the change in the quantity demanded through the change in the price and this change will be decided by the indirect utility function