 And our topic is the budget minimization subject to the utility constraint. It means now that mostly we have studied up till now that the consumer will maximize the utility given to his budget constraint. But this is not always the situation because sometimes consumers want to increase the utility but he knows that he is having a limited amount of the budget in his hand. So when he is having a limited amount of the budget, sometimes he is going to decide that rather by increasing the amount of the utility, he decides that I can reduce my budget in any way. I mean whatever I have to attain, I should keep my utility level constant and I should keep this constant level of utility. So it is just the reverse of our previous that we mostly have decided that the consumer when they want to maximize their utility, we consider that they will maximize their utility but their budget that will remain constant. Here we will assume that they are going to attain a particular level of utility and then that particular level of the utility is kept constant. Now the consumer has to optimize that how he can attain a minimal budget to attain that given amount of the utility. So it is as the similar nature of the graph that we have shown in our various previous lectures but here we can see that the consumer as previously we were having that the consumer is having one budget line, here we see that the consumer is having this one desired level of the utility that now this utility level will be consumer's constraint that in any form consumer has to attain this utility. Now what is the possibility? So the consumer has to check that either this budget line, either this budget line, either this budget line whatever the budget is available. So by adjusting his budget or by optimizing his budget now the consumer has to attain this utility level. So now the possible budget lines if we see that now there are the three budget lines, this E1, the second is E2 and the third is E3. As these are the budget line but for the consumer we can say that the consumer when spent all his budget for the attainment of the utility this will be the expenditure of the consumer. So here now we have to assess that what should be the optimal parts of the price and the commodity quantity that the consumer will select by the expenditure that what will be the required level of the expenditure that will keep the consumer on his existing or attained level of the utility. So expenditure level when we see that it is tangent to our already attained utility level this point will give us the equilibrium point and as we were dealing with the previous approach that where we were utilizing the maximization of utility that at this point slope of expenditure line is equal to the slope of utility function. So equality of these two will provide the opportunity to the consumer to decide that that is the optimal point for the consumer. So if now we utilize the mathematical derivation we will just shift that now this is the expenditure that was just equal here income was equal to price of x, x and price of x2, x2 plus price of xn and n and when the consumer is going to spend all of his income for the purchase of this then it comes in this form that it is the expenditure. Now consumer has to optimize his this expenditure subject that he is going to contain his utility constant and through the lagrange now the objective function will be this and we can now give the this income equal to e minus p1 x1 minus x2 and minus pxn xn plus lambda into our subjective function that is the utility. So with the help of the Lagrange now we can come up with all the optimal points of the expenditure and at the end what we are going to come up the result it is the similar as it was of the utility maximization. So our problem that how the consumer has to increase his utility with the given budget or either the consumer has to minimize his expenditure with the given level of the utility by the both ways we come up with the same conclusions that the consumer's expenditure will change by only two factors if the consumer will have to face the change in the price or if the consumer has to face the change in the income. So in either way consumer can come up with the same type of the decisions and with the same type of the results where the lambda will be equal to the change in the commodity demanded divide by its respective price. Thank you.