 Chapter is consumer behavior and today we have to study that what are the properties of the Hixian demand function. As we have already studied that Hixian demand function, how it performs and how it is derived. But when it is derived, it cannot be possible that if it is not having certain properties and it is not going to fulfill those properties, it will not exhibit that function that we have expressed during our derivation. So now we are coming to its various properties. The first of all, the first property is that there will not be any excess utility for any commodity of X and that X when it is the member of the Hixian demand function and that Hixian demand function that is depending upon the price and the utility kept constant. So in this Hixian demand function, price for that commodity that is under consideration will only vary but the utility will be kept constant and the prices of all other goods it also will be kept constant. When we come to the second property, it means that property it follows from only one concept and that concept is that the utility function is continuous. If the utility function is not the continuous, that will not be possible for us to have the substitution effect because the consumer is going to substitute the amount demanded at one price to the other price but by having his travel on the same consumer demand function or on the same utility function. So the continuity of the utility function is the basic essence to have the substitution effect. And when we come to the third property, it says that we can assume that if there is one any commodity of X that is the member of one Hixian demand function and that is so that that utility of that X is greater than previous utility U dot. So if we assume this, then we have to assume another bundle and that is under the property of the fourth and there here now that commodity that is the X prime that again it will also be the member of that previous bundle of the X and here we can say that the one scalar vacuum should be there and that is not a vector. That scalar will just denote any amount that it will denote any number. So this scalar, it is the member of that number bundle and it can approach from zero and it can go up to one. So when there will be this scalar number and by the continuous process we see that when we select then the consumer can approach its utility and it can move maximum and maximum to up to one. So when it approaches near or close to one then we can have that utility for that X prime it should be greater than the previous one utility of the that we were having. If this property prevails there then the price attached with that bundle that is X prime value. Now that price multiplied by that amount it will give the expenditure function. So here if utility is greater then we can say price into X prime it should be less than the price into X one and it will contradict the previous one that being the optimal in the expenditure minimization problem it cannot be possible that now we are going to have the same utility level but we are going to have the more expenditure because this property is must for the continuity of the utility and even for the Hixian demand function where we keep on this thing that the nominal income it will remain the same but the real income it is going to vary somewhere. So for that we require certain compensation and when there will be compensation then either this compensation will be negative either this compensation can be positive for which the consumer has to be compensated through certain either addition or by a deduction in his nominal income just to keep him on the same real utility level on the same utility level and for this this property has to be fulfilled. One another thing that we are going to see that is number one its property one another that is there should be convexity and uniqueness and when we say that convexity our utility function it should be convex and it should if it is not convex or rather it will be in the form of touch and cave it will be very difficult to express all the properties that a Hixian or a normal even demand function it's going to have. So if that function is strictly convex then we can have the utility and we are expressing this by the utility dot mean this is the continuous utility function so it is strictly cause icon cave and this property it tells that there should be a very very particular unique element in the Hixian demand function and if that utility function if it is expressed for more than two commodities mean commodities X1 and commodity X2 then we can have that this utility function is just equal to alpha natural log X1 plus 1 minus alpha into natural log X2 it means here the scalar value that is attached with that utility function it should be equal to one unity when it is added up so when we will add up this alpha and 1 minus alpha it means the total scalar value is equal to 1 and when this continuity is differentiated with respect to the various aspects of the variable proportions of the commodities mean it can be differentiated with respect to either X1 or it can be differentiated even with the respect to the X2 then the optimal consumption bundle it should be characterizing only having through the first order condition and that first order condition it will give us a very peculiar property that whenever we are going to derive that the output that we are having in our hand it is the same as we are going to deal with one commodity or with the two commodities so Hixian demand functions properties are very very necessary to be fulfilled otherwise Hixian demand function will not be possible to execute all the functions that have been expressed through its derivations and all the other problems Thank you