 Under the topic of consumer behavior, we are going to study the assumptions of consumer preference ordering and the assumption difference stability we are going to study today. This property allows that the consumption function or the any function because this property is derived from the mathematical derivation. This property says that any function that has to be measured, so the rate of change or the slope of the function of that any point can be measured that will give us the property of differentiation. And if we say in other words, any function or graph that we have to calculate the rate of change, so until there is no property of differentiation in it, we cannot apply that rule on that. Now, if we look at differentiation, we call differentiation as derivation in mathematical terms. Now, for derivation, the convexity of our curve, the continuousness of both the properties will have the property of differentiation, then we will be able to apply the rules of optimization. Now, if we look at this property on our consumption functions or utility functions, then if we say that this property tells us that the utility functions can be differentiated and that differentiation will be the required order for which we want to do it and this assumption also tells us that if we are applying this property under this, then the indifference curve or its surface will not have a sudden jump in it or if we look at it, the gap will not come and the sudden jump will not come, then if it is downward sloping, then it has to be regularly downward sloping and if it is upward or positive sloping, then it will go in the same form and if there are jumps or upward movements, then its slope will change, it will not remain the same. That is why when we have to calculate the required rate of change or we have to do the compensation, then we have to calculate the inward movement of indifference curve along with it. And that is why we have the important examples in which we mean real analysis in mathematics, that it is a branch, if we calculate it in the same way, then we apply the derivative and we apply it, we are applying the method of Taylor series, we are taking optimization techniques in it, so the basic sense of all these functions is that we assume the possibility of the function differentiation, then only it is possible, if we want to explain this property mathematically or graphically, then we can say that any curve that is explained on the x-axis and at the same time the other commodity that is explained on this y-axis and the combination of these x1 and x2 properties, if we draw two lines and here we keep the quantities of the different commodities, I keep the quantities and make a combination that we say that here x1 was 1, here 2, here 3, here 4, so in this way we have various commodities and in the same way we say that here we have y1, here 2, here 3, here 4, so the combination that we will make in front of us, so when we see the commodity of x1, then we have the commodity of x2 in maximum amount, so if we see from here, then one commodity is increasing in this form, suppose we say that it goes up to 20 and in the same way the 20 of x2, if we start from here, then in the same way it will come downward in the same way, now the various combinations of these, now if we plot them in the form of this curve and basically if we see these combinations of points which we draw that this was one point, this is one point, these points we do not draw on the curve, rather we have already kept all the combinations points on our graph and after that the curve is basically that line that is going to join all these combinations of our preferences or the selection models or we can say the various combinations which the consumer has selected, if we see here then this consumption bundle can be A, this is B, this is C, this is like this and now the budget line of the consumer is constrained, if we see under that and we draw this constraint, then where this equilibrium point is given, then we say on that curve, then this slope, the tendency of this line to this, this to this and this, the rate of change we are calculating means this rate of change is change in X2 with respect to change in X1, I mean if we take one unit of X1 commodity, then for that we have to sacrifice the commodity of X2 and similarly if we see it going backward, then we can say that if we go along with its movement, then we say that the commodity will have to sacrifice the commodity of X1, so in that form if we see, then we will change in X1 with respect to X2, then this slope and if we have to do this curve, then until in our function which we cannot calculate the very small amount of rate of change, which gives us the property of differentiation, then we cannot solve this function, so this property is the basic essence, if we see it in this way, then the ratios that we have have come to us, then it means that these ratios can be in a bigger amount, if we go in this curve, then if we see it now, then this amount is a good characteristic and similarly this can also happen and this too, so in optimization technique, if we see it, then until here it has been decided that we have to calculate the rate of change and there will be various amounts for it, now how much these amounts should be, for that if we see, then the mathematical rule that comes to us, that change in X2 with respect to change in X1, keeping in view when utility of the consumer is kept constant and when we calculate it, we now calculate its limit and limit mathematical tool tells us that it is a small point that not equal to zero but change in X1 is approaching to zero, i.e. we say that change in X1 is approaching to zero, so this means that the same curve that we have calculated here, so if I clear this one change, that this was such a line and for this we had this one part, so rather if we say it, then if I say it in a magnifying form, then we can say that actually this part, this is the part of the curve, then if I look at it from a minor angle, which we understand that if we are looking at something through a lens, then if we see it in this form, then we can magnify it and see, so in this way, this is a very small point and in this way, if we look further, then even smaller than this, it means that slope is basically measured with respect to to change in X1, when this change in X1 was approaching to zero, that if we just do a little more than zero, then we will say that this is coming to us in that form and when we will do this movement, we will see the points in the left word in the difference curve, then we will see the derivative, then its absolute value increases, now if we look at the increase form, then from here when we go to the downward form, then in this we will see that our denominator was X1 and our numerator is the change in X2, while changing from here, it is moving from here to here and this rate of change, if we calculate this, then basically, this is our substitution of X2 with respect to X1, then the substitution of this rate of change is called the rate of substitution, so if we look at this rate of substitution, then if we say that this is the marginal rate of substitution of good 1 with 2, good 2 comes to us, then we explain it in that form, in which we say that if we calculate the change in X1 with respect to X2, and in the second form, if we did one, then we did change in X2 with respect to change in X1, now if we look at this, then with this change, the negative sign that we show, then this negative sign is actually explaining the slope of this curve, that with the passage of time, now when we go to this change, then if we look at the denominator in the change, then this is X1 because the change increases and X2 decreases, so over the time this change is becoming a declining show, and if we calculate this in the upward movement, then in the same form, the numerator is getting smaller and the denominator is getting bigger, then the negative sign is showing, so this negative sign is showing the slope of this change, and when we say that when we say keeping utility constant, this means that the function or curve along, we have to calculate it, our basic assumption is that utility of that will remain constant, so if we do not have a constant in the form of utility, then we will not be able to include the property of its substitution, and therefore now if we look at this, then utility of one commodity that is expressed by this, and similarly utility which we calculated in the same way through the other slope, and if we divide these two together, then our marginal rate of substitution is explained in front of us, so basically if we look at the marginal rate of substitution which we are calculating, if we look at of commodity one to two, this basically is the rate of change or the marginal utilities of commodity x one with respect to commodity x two, so this will give the further explanation of the consumption function bundles utilization and the consumer behavior.