 Utility maximization subject to the budget constraint under the chapter of consumer behavior. Utility maximization is required when we have to optimize the consumption of a consumer. And this can be possible that the consumer is going to maximize the utility without consideration of the budget or with consideration of the budget. As we know the consumer has various preferences available to him and at the same time consumer has a very long list of the wishes to attain various commodities. And as per the law of the monotonicity we can say the consumer is always willing to have more and more and more. Then what is the end? Either it is possible that the consumer is going to attain more and more without any cut in the real life, we know that this is not possible because if consumer is have many and many wishes then there are the limited resources. So these are the limited resources that provide the limit to these utility preferences or the wishes. So now we want to assess that from the available bundles which bundle is going to provide the maximum utility to the consumer with the available resources that he is having on his hand. So from the graph we can see that here we have plotted 3 indifference curves and these indifference curves they are in the order of preferring that this indifference curve is having this is if I will say this with the 3 that it is the highest level this indifference curve 2 is less than 3 but it is having more commodity or the more rank that the indifference curve 1. So these are the available indifference curve and consumer will be willing to have either this either this or either this but mostly he will prefer the more. Now the consumer has to decide for the bundle of the goods with the highest indifference curve that the consumer will prefer. So in the diagram we can see that we have drawn this line that is the budget line and at this point we see that this point of equilibrium has been attained when the consumer is having the commodity X1 on the expected and with this much amount of the X2. If we see the consumer if coming to the higher level of the indifference curve he may have this point or at the same time if we draw the lowest he may have this point. So if I draw these points so point A, point B and the C but out of these indifference curves the consumer is going to select the bundle B because this bundle B is not only available on his indifference curve but because it is also within the affordable range of the consumer. So this affordability that is the constraint provided by the budget constraint so for utility maximization consumer will prefer that bundle that will give the maximum utility to him if it is available within his resources. So for this we can say now that at this point this is the indifference curve and we can have what is its slope and this budget line it also has its slope. So when the slope of these two curves it will become tangent to each other so we can say the slope of indifference curve is equal to the slope of budget line. This will give us the total point of tangency and this will provide the consumer with the maximum utility. If we look at it in the situations of real life then the consumer has a lot of requirements but if we look at which thing I have to buy then he will always assess that the things or the bundles of its price and market that comes in the range of its income and its budget so when we will assess its maximum utility for that time then we will have to see that the affordability of its budget will be present in its utility for that time and its maximum utility will be given. Now because we are dealing with it in real life so we have to first make some parameters here so that we can see that the bundles available to him we have to do our focus for the measurement only to him number two we have to see what the market price is at that time and only counting the prices of that time we have to draw the budget line and number three we have to draw the slope for these two things and when we attain the highest level of indifference curve then we have to stop for the purchase that now the maximum level has been attained. But only the graphical representation it is not possible to have the solution of the various thing in the economics we may have not only the graphical but we have to draw the utility maximum point with respect to the mathematical derivation and here we see that there is a one bundle and for this we can say that commodity x1 and the commodity x2 and our objective function is to maximize this utility and here the subject or the constraint that is having that is the two commodities pricing and with their available commodities quantities and it is the budget line provided by this so now there is the possibility that we can draw this by the two methods one by the substitution method and the other through the Lagrange multiplier so when we utilize the constraint method from the budget constraint we can draw the value of x2 and from their same we can draw the value of the x1 by substituting the value of x1 and x2 in our utility bundle we can find out the value that is going to provide the maximum utility to the consumer and according to this if we are going to utilize the Lagrange method we can utilize again the method by this that it is our objective function and this is our subject of the budget but here we will not utilize the values of the x1 or the x2 for the substitution rather we will derive a Lagrange and that Lagrange we can say that by this method we are going to draw now that we have to maximize this Lagrange L where this is our objective function and this is our subjective function with this lambda and for this we can now take the marginal utilities of the function mean marginal utility of the Lagrange one with respect to the price of with respect to x1 and the second with respect to x2 and so we can assess by this method that what will be the level for this change in Lagrange with respect to change in x1 change in Lagrange with respect to x2 and then change in Lagrange with respect to lambda so with respect to these three variables we will take the first derivative of our Lagrange equation and after this now we can equate all these three equations in a way that we get the value of lambda on one side so when we will have the value of lambda then we can equate one equation with the another and here now we can get the ratio of these commodities with their prices so we say that price of one and price of two when multiplied by their respective commodities it becomes the expenditure on that commodity so in this manner we can say that we will attain a point where the expenditure on the good one will be equal to the expenditure on the other commodity and here we see by this function that the expenditure on good two is the double than the commodity one it means that we are spending more on the commodity two by substituting the values in equation three now we get the amount of the x2 or we can say that this was our optimal level of the commodity x2 that we can have and from this equation again we can calculate the by substituting the amount of x1 so we can say by rearranging these equations through the Lagrange we can assess that what was the amount of the income or what was the amount or the portion of the budget that the consumer is going to spend on each commodity and this amount that we can have from one to the other we assess that how the marginal utility of the commodity vary with the change in the commodity's consumption