 that is, what are the properties of the indirect utility function? As we generally know that mostly we derive certain decisions or draw certain conclusions related to any type of the topic and this conclusion that we mostly infer from various mathematical derivations for this we have to assume certain parameters for that function and for this there should be certain properties so if we draw some conclusions from the indirect utility function how it will behave and what will be its impact then it is necessary that some assumptions or properties that we have fixed will be fulfilled so to have our general conclusions of some of the indirect utility functions properties that we should have which we have the first property that is the continuous function and no illusion or what we say the money illusion it means that the indirect utility function that is the homogeneous in the form of the amount of the or any type of the degree of the M or the money income and if there will be increase in income and direct utility function will be increasing and for the prices indirect utility function will have the decreasing response and it must have a quasi-concave function and then the rise identity now we will discuss all these properties one by one so our first property that is called the continuous it means that the utility that the consumer is going to express it must be in the form of the a continuous function that there should not be any type of the stoppage or the gap means consumer will perform the any change in the utility if there will be any change in the price or if there will be any change in the income then the consumer will exhibit his preference on a continuous form of the utility function and on that utility function he is having any adjustment process moving from one part to the other it means that property of the continuity it ensures the movement or the response of the consumer to the changes in the income or to the changes in the price now coming to the other property we see that there is the no money illusion what we mean no money illusion it means mostly when we take the utility function of the consumer we take that utility is the function of the price and the money income or the income provided to the consumer here to prove this money illusion we have added this parameter or the coefficient to all our these variables so if we take that what will be the change in the optimal level of the utility due to change in the money income then we by taking the first order derivative we see that the change in the utility through the change in the commodity x quantity and how the change in this money income has caused me in the change in the purchasing power has caused the change in the optimal level of the x so plus this will be continue for this amount of good and for this amount of good if it will be multiple good case then we can continue this in the form of x y z and like this or x 1 x 2 and x 3 so it means that this will be the total change in the expenditure of the consumer incurred for the purchase of the commodity x and this will be the change in the expenditure incurred by the consumer on the commodity y multiplied by the lambda and when we come up with like this this part and this through the property of adding up it comes to all this price x n multiplied by their amount of x n divided by m this comes to total expenditure equal to the m so we come up with the lambda it means whatever level of k we are going to add here it means if we add by the 2 or we add by the 3 or we add by the 4 so whatever there will be the amount that will be cancelled out by the change in their money income or adjustment so now the next property third that increasing in the income and decreasing in the price form we are going to describe this in the form of a combined factor as we know that mostly in the form of the utility there is a normal behavior that whenever the income will increase consumer will shift its budget line to the right side and when the budget line will shift it provide the opportunity to the consumer to acquire higher level of the utility function so that the consumer will purchase more of all the goods that he was having in his previous budget bond but at the same time if there will be any change in the prices consumer will respond in the form of the opposite means if prices will increase then consumer will reduce its utility if prices will decrease consumer will respond through the increase in its other form so there will be the response in negative in response to the prices because the consumer has to adjust accordingly now the next property that the indirect utility function expresses that is it is quasi convex in the its formation as we see that the utility function they will be mostly in the form that they will having this shape provided that the consumer will be able to have the adjustment of all the bundles on the same level that is the property of continuity and if it is like this if there will not be this type of the shape it will not be possible for the consumer to have the tangency with its available budget line so to have the tangency condition fulfillment we have to ensure that the indirect utility function they must be shown in the form of the quasi convex form so this tangency point we can have with the amount of the m or the expenditure like this and as consumer increases the utility of the consumption of the commodity so in the similar manner respective expenditure incurred on that utility will be increasing so as the utility in the form of direct marshalian functionality increases in the similar manner the expenditure incurred on that attained utility it will increase on the right side form the sixth part is or the property that is called the rise identity so when we say the rise identity it means the consumers marshalian demand function for any good it is none other that it is the ratio of the partial derivative of the indirect utility function of the consumer so when we say that this is the optimal part of the consumer utility and when we through envelope theorem we measure the change in this so the change in the utility due to change in the price then we come up with this form and it say that it is none than other the change in the expenditure through the change in the price so according to this we come up with the part that is called the amount of the lambda and lambda is none than other that is the one unit change in the price of a commodity how it will have the effect on the change in the expenditure incurred on that commodity so it provides the consumer a tool to compare various commodities or we can say the law of EQ marginal utility through this property