 That is the derivation of the Hixian demand fund. As we have already discussed that the Hixian demand function is the relationship of the quantity demanded of the consumer with its change in the price keeping the utility level constant. So, if we are having the utility constant, it means now we can derive the same demand curve by the two methods and here we have to assume again our previous notations that there are the two commodities and with the two commodities they are respective prices and now the price of one commodity is going to change but the other good prices that will be kept constant and here the utility will be kept constant and the factor of income it may vary from different point to different point. So, when we deal like this, we can derive that now the keeping the utility constant we can derive that curve once on a graphical form and on the other form we can derive this on the mathematical to have a more representation. So, when we explain the Hixian demand function on the form of the graph we express this on the two dimension on which there will be on x-axis one commodity and the other commodity for which the price will be kept constant that will be on y-axis. So, we will express the change in the quantity demanded of a single commodity with the change its own price. So, coming to the this graph, we will explain that if there is a one utility curve or the indifference curve that the consumer is having, so if it is like this on the this curve we say that the consumer is having the opportunity to select various consumption bundle like this but already we have shown only three and these all consumption bundles are like this that the consumer is indifferent among all this. So, consumer can select any from all these. So, along this the same indifference curve if we take only three that I am going to write here bundle A, bundle B and the bundle C. So, for these three bundles if we draw the tangent to these points we can see that there will be difference in the slope of these tangent line. So, any line that is tangent to bundle A it will have a steeper slope, any line that is tangent to point or bundle B that will have somewhat flatter slope and then a budget line that is tangent to point C that is farthest flatter than the point B. So, it means there is the change in the slope from upward to downward like this and the slope is going to reduce in on the right side form. It means when the price of a commodity it reduces then we can say the slope is decreased on towards the right. But on this curve we can if measure one and other budget line for which we are going to compensate the consumer for its change in the expenditure or the change in the budget. So, if we explain our this change in the quantity demanded along a utility curve we can see that in this diagram consumer is having one utility curve and on this utility curve consumer can have the opportunity to avail various bundles and for all these bundles consumer is indifferent to each other and consumer it means he is having no preference for any other thing. But now who will select we can see that suppose this is his bundle A, this is his bundle B and this is his bundle C. So, then we see that in three points against if we draw tangents like this then we will see that this point A the line which has shown as a tendency its slope is steeper and the next line is somewhat flatter in comparison to the first line means its slope is less and at the point C this curve is more flatter so if we show it in this form then it means it will be like this and other way we will say its slope is going to be less. Now, since this slope is actually ratio of price of X and price of Y so in these two slopes we see that the price which is at point C which we said if I call it price of XC it is actually less than the price of that that is point B and this is less than the price at the point A. So, as the price of our numerator is less then we see that our slope is less and the curve is flatter but if I explain this point through Hixion then we will see that on Hixion overall if I show these two green lines here then that is their slope in black lines is slightly more so this is flatter so it means this green which is here is showing the Marshallian demand curve so this black is the Hixion having steeper slope and in this comparison if we say on B we say that both these slopes were equal but if we go to this point so Hixion will be steeper than Marshallian and when it is flatter then the response of Marshallian is more so the Marshallian will be up and Hixion will be downward. Now it means the slope's difference is showing that actually the change in price which is the consumer's expenditure may change if we compensate him or if we look at the compensation here we see that it is Df and it is the consumption in which we keep the utility constant so this is our Hixion so we see that with the change in the price either decrease or the increase just change in the expenditure required to compensate the consumer if we provide the compensation so it will show the similar nature of the direction but with the less magnitude and with the less slope so the Hixion curve it will be steeper than the Marshallian demand curve because it only and only includes substitution effect rather Marshallian it includes substitution effect plus income effect and through compensation we have compensated for this income effect to the Hixion demand function and totally if I explain this in the other diagram so here if consumption if we look at compensated or uncompensated so when we compare their price changes we showed the shift on the right after the original budget line and then the compensation we showed on the same budget line so this is the movement which is shown by our compensation point e1 or e2 and e3 is the final so this shift of e1 to e2 and to e3 when we say so shift of e1 to e3 is basically the Marshallian curve because it includes substitution effect plus income effect but because Hixion demand has to deal with the substitution effect so its movement will be from point e1 to e2 and this will be our compensated or Hixion demand curve