 And, here we are going to study the corner solution, corner solution is a very unique type of the consumption decision in which consumer is going to decide related to the consumption of a bundle, but in that consumption bundle consumer is having a very special type of the preference that he is going to prefer only one commodity than the other things or we can say as compared to or we can compare with the budget line if he is spending all of his income for the purchase of only one thing than the other things. So, in this way, consumer will not be having any ability or consumer is not having any willingness to have trade off of the other things. So, the consumer as he is having the preference only for one commodity. So, his utility increases only with the purchase of that commodity in any amount. So, if a consumer is having two types of the commodities X1 and X2 and consumer is having the particular type of preference for X1 to how much amount of the income he is having or whatever is the price of the commodity, he will go on purchasing only X1 from one indifference curve to the higher level of indifference curve or any other in any type of the decision he will prefer to have only that particular commodity. So, when coming to the diagram, we can draw here various types of the corner solution. So, if consumer is having two commodities and suppose he is having this budget line and now consumer is having the preference only for X1 in this way, now consumer will have indifference curve in this sense that it will intersect the budget line only at this point of intersection where the whole of the budget is only spent for the purchase of commodity X1. So, if now consumer budget is increased that does not mean now consumer will purchase some amount of X2, he again will utilize his whole budget to purchase only and only this next amount of X1. So, in this form the consumer may have various type of the indifference curve but they are intersecting with the budget line at the right side of the axis. This shape was possible when the consumer was having the indifference curve for two commodities which may have the like the shape but the consumer may have very straight line of indifference curve and for this consumer can have like this. Now the consumer may having a preference for the commodity X2 and if this is the budget line now consumer is having indifference curve in this shape that they are going to intersect vertical intercept and with the budget line at a point when the consumer is spending whole of his income only for the purchase of commodity X2 and if he is again having more income he again is going to spend all of his income on this purchase of X2. So, whatever is the level of indifference curve he is going to increase his indifference or the preference but only through the purchase of one commodity and not having the trade off with the other commodities. Again we are facing the same situation of the consumption decision so consumer if has to maximize his utility and hypothetically he may have one to two or the other commodities available to him and he is having certain amount of income so but now however there is one assumption where we say that that optimal point for which the consumer is having highest level of satisfaction or a point of satiation for that the assumption that that it should be greater than zero but that assumption is dropped when we are dealing with the corner solution because when we are having the corner solution one commodity will be having the amount of utilization only zero. Now coming to the point because when the consumer will not be able to have the trade off so he will not be having the trade off so there will not be the possibility to having the slope which we have talked about earlier that in interior solutions where we equate the slope of the indifference curve and its budget line slope graphically we say that the consumer has attained equilibrium point but because in the corner solution its slope will not be able to be made because it is available at the same point of extreme and if we look at the same extreme point then now the slope of indifference curve is not available so how should we derive its equilibrium point or its consumption decision so here we help out algebra or the mathematics we help out and we can solve our this equation not graphically but with the help of the Lagrange and when we utilize the Lagrange we derive when Lagrange is equal to the our objective function of the utility subject to the budget and keeping in view the marginal rates we derive our first derivative that change in Lagrange due to the change in x1 and when we derive up to this point we utilize these points 1 the 2 when extreme or the optimal points mean this is derived and like this is derived and then we are coming to this decision where we can have that the change in utility due to the change in price with respect to the optimal point of x1 it can be equal to lambda and when lambda is taken on the other side that can be equal or the inside this can give you and so here the lambda it will give us the marginal rate of that commodity per unit price so graphically it is difficult to explore or decide the consumption of a consumer for the corner solution but with the help of the Lagrange it is possible to derive the consumption decision of a consumer pertaining to the that commodity for which the consumer is having the whole preference than the other so now if we say it in economics form that a consumer when he spends all his income for one commodity and for the other he is not ready to exchange himself so we can say that the consumer he spent all his income from the consumption of only one commodity