 And, we are going to study the topic of Hixian demand function and what is the concept? Hixian demand function tells the relationship between the quantity demanded by the consumer with its price keeping the utility constant. And the concept is somewhat different from our previous demand function or the demand curve that was explained by the Marshall. In that curve or the function we kept the money income constant and here we will keep the utility constant. So, the basic question that the consumer is facing is what is the level of the utility, how to obtain the utility and how to optimize the utility when the question of optimization is faced definitely there is the resource constraint. So, the problem of utility maximization keeping in view the resource constraint is solved in the two way that is called the consumption duality or the duality principle means that is the mirror image of the each other. We can say that in the Marshallian approach that we tell this approach by the name of the primal demand curve or if I say that the Marshallian method we say that this is basic primal the basis in which we change the value of commodity and keep its income constant and keep the value of other goods constant. So, in the second method that is called dual or the other the mirror image of the primal is that here we are going to maximize the utility and this procedure maximization of utility is achieved not through the utility rather through the cost minimization. So here the expenditure incurred by the consumer that can be minimized in a manner that will ensure the existing or the cap utility level of the consumer constant. So, in the primal the thing that will vary that will be the price of that commodity for which we are going to measure the quantity demanded and there will be the change in the real income but in the dual Hixian demand model utility will be kept constant and if utility has to kept constant it means something other has to vary and that will be the nominal income. So, this Hixian demand curve that was basically provided by a British economist and so it is on the name of Sir John Hicks and Sir John Hicks actually conveyed that in the relationship of quantity demanded with its own price changes if we are able to compensate the consumer for its change in expenditure or for its change in real income then we can draw a relationship. So, for this we assume that the utility of the consumer it will be kept constant and for that constant level of utility what will be the requirement in the expenditure that will be compensated and that amount of compensation can be calculated through the compensation variation principles and by this we can draw the Hixian demand curve. So, when we say that there is a demand curve and in Marshallian we say the demand curve of the x or the demand curve of the y or the demand curve of the z it depends upon price vectors and where the price vectors are the price of own commodity and the price of all other commodities. In Marshallian approach we will keep the vectors of the prices and the income of the consumer constant but in Hixian we will assume that the demand of the consumer function is the function of the prices and keeping the utility level constant. So, when we consider this and we draw in the form of the equation in the form of comparison we can have that the same level of the factors we are having here the price of x and price of y and on the right side we are having this Marshallian or the I will say them the primal demand curve and here it is the dual or the Hixian demand function. The only difference is that in primal we will kept the income constant and in the dual or the Hixian we will keep the utility level constant. So, dealing in the both we are having only one objective that we want to examine or we want to assess that the demand of any consumer for one commodity can be changed by changing its price. So, the same method we utilized of the Lagrange principle and we come up that the change in the demand due to change in price and the change in demand due to price in the primal. When we differentiate these functions and the first order conditions we equate to each other then we come up with the one part that the Hixian demand curve it can be equated to the Marshallian demand curve and in the Marshallian we can substitute the amount or the part of the income with this of the expenditure. Now these two demand functions they become equal and they become equal in a sense that this will tell us the amount of the income or the amount of the expenditure that will be particularly required when the consumer has to maintain its existing utility level. So, when we have to draw in the form of the curve we can see that this is our general Marshallian or the original demand curve or the ordinary that we have discussed many times and this is the center point and here I can see that the consumer is at equilibrium and this is the center point and if now we say that there is a point when the price of the commodity X it is more than that the previous so we say that the consumer shifts backward and he reduces his demand and if the price of X reduces then in response to decline the consumer increase its demand and it moves rightward. But when this movement we have to show on the form or in the form of Hixian demand curve the direction of the change of the quantity demanded will be same but the magnitude it might be different and this magnitude it actually expresses that in Marshallian the effect of the change in the quantity demanded is due to income effect and the substitution effect but in Hixian because we utilize the concept of compensation how we are at the point of compensation we say that if this point where I say that if it is the point of A at this point commodity that the price is the price of X1 that is higher than the price of X which is X2 or responsive of point B so when the commodity's value increases then we will say that the real income or its purchasing power will reduce. So for that reduction in purchasing power now the consumer it requires some amount of money in his hand and that amount of money is now extra required to that extra required money and that extra required income we will say that that is the positive compensation because we have to add the consumer why do we have to do it because we want that the utility level of the consumer that was present earlier should be maintained means the amount of consumption that was purchasing earlier should be purchased so in this way we compensate the consumer so when we compensate the consumer the income effect that was on the marshalian demand we compensated it and now the only change in the quantity demanded that will be only left with the substitution effect and likewise if this decrease in the income case decrease in sorry price case in which it has an extra amount of income left so we will take this amount away from it again we will have income consumption it means that will be the negative income consumption so it means the amount of nominal income M that will vary in the three cases once we can give one we can take from them so keeping in view this we can say that this curve because the equilibrium points are showing our tendency when it is equal to slope of the budget line so because here the price is changing on three points and but the seven compensations are added so because of that our compensated or Hixian demand curve will always be steeper than the marshalian means its slope will be less because it is