 Today's topic is how we are going to derive the indirect utility function. As we have already discussed that the indirect utility function is a technique or a method that we utilize in the Marshallian demand function to assess the change in the utility of a consumer via change in the price and then we calculate that this can be described by the change in the price in the form that we have to measure the change in the expenditure of the consumer with the change in his price. So when we calculate we utilize the mathematical technique of the envelope theorem here and we say that if there is the utility curve of the consumer that we have mostly utilized in the form that utility is the function of x1, x2, x3 and up to xn and for this the consumer is having the respective prices in this form and now we have to calculate the utility of the consumer we mostly take the utility maximization approach and when we take the change in the utility due to the change in the price we can utilize this price and x in this form that utility is indirectly equal to the expenditure incur on the price of this commodity. So expenditure incurred on commodity x will be this expenditure on the commodity of x2 will be like this and the expenditure on commodity 3 like this and like this the commodity n and when we add up all these we can say that it is equal to summation of all the prices multiplied by their respective optimal bundle of the commodity. So this will be our optimal or the desired budget or the expenditure. Now the utility is maximized through the optimal level of the budget or the optimal expenditure that is required to provide our desired level of the commodity and what will be the change in our optimal utility that will be measured through the change in the optimal required budget. So it gives us this equation that we can measure through the indirect utility and when we measure through change in expenditure with respect to price we come up here that it can be equal to this that when we sum up this will be equal to the summation of all the utilities and change in the optimal amount due to the change in the price. So keeping the all Lagrange equation that will be equal to this part and we say now this is the change in the utility due to the change in the price and this can be through the change in the expenditure and it tells us that the budget constraint of the consumer it should be same or the nominal income it should be kept constant. So we can measure the utility not only due to the change in the price through the utility rather through the change in the optimal amount of the expenditure or the budget required to attain that amount of the utility. So when it is ensured that this part that all the price and their respective expenditure and this any change in the optimal point that will occur due to the change in price it will give the optimal point of the same commodity. So it tells us that any increase in the price of this commodity that will cause the reduction in the purchasing power of the consumers it will require from the consumer to reduce his income or to reduce his expenditure incurred or that commodity. So that now the consumer will be able to at least purchase his previous bundle and at the same time he will have certain amount in his hand now to allocate equally to the all other goods. So by Shaffer lemma and utilizing the first order conditions now we will say the purchasing power of the commodity it will reduce and that purchasing power of that will reduce it will provide the consumer a further availability in the form of certain real income that now consumer will incur for the purchase of the other goods provided that the goods that he is going to purchase they are mostly the normal goods.