 In consumer behavior, today we are going to study the duality between the Hixian and the Marshallian demand curve. Hixian and the Marshallian demand curve we have studied already and as per the Marshallian and the Hixian both approaches, we can say when there is increase or the decrease in the utility due to the change in the prices, we are having almost the same behavior but with certain restriction that we have to impose. In Marshallian demand function, we utilize the two commodities X1 and X2 and we assume that the income it will remain same. But in the Hixian demand function, we again utilize the two commodities X1 and X2 and for those bundle of goods, if the utility has to be maximized, there will be one assumption that the utility it remains same and utility remains same but the income it varies and this income that varies that is the nominal income. And now coming to that, we have the different equations for both Marshallian and the Hixian demand curve. When we explain this from a graphical presentation, we can see that on the X axis, we have the same one commodity that is quantity of X and on Y axis, we have put its price. Then what is the change in the price and that it is going to have certain impact on the change in the commodity demanded. We see that there is one Marshallian demand curve that we have explained by these points that shows the three points of the quantity demanded, this one and this the central point of that is shown by a point that is just equal to the price of X with double prime and it shows the quantity demanded X double prime. And if I put the point one here, it means that will be our one initial point. Why? Because when there will be change in the price in the sense that the respective next price is lower. So, when the X price will decrease, we will move towards right and when there is increase in the X, we show this by a movement on left side. So, whenever there is decrease in the price, we see that the consumer has shifted on from this point one to point two on the Marshallian indifference curve. But when we see on the Hixian demand curve, the consumer has also shifted from one point to this point that is two prime, but here the consumer is having less quantity demanded. It means here the consumer has both way increased its demand, but the increase in the Hixian form is lower than the Marshallian. So, now when the other case, consumer is having a situation when the price of X increases. In the incremental form, the consumer shifts from this point one to left, but in Marshallian demand curve, he moves farther than the Hixian. It means now for the Marshallian demand curve, the consumer has reduction in its quantity demanded and also for the Hixian demand curve, he has also reduction in quantity demanded. But this decrease is more in Marshallian demand curve. This shows that in both cases if we compare, the Marshallian demand curve is more responsive to any change in the price as compared to the Hixian demand curve. Now, coming to that point that either that will be possible to achieve all our desired points if the consumer is having either Hixian demand curve or the Marshallian. So, according to the duality theorem, we can conclude that either the consumer is on Hixian demand curve or either the consumer is on Marshallian demand curve, then there is a situation that we can have an intersection points of both and we can calculate the quantity demanded of X. Either we are having compensated demand curve or either we are having uncompensated demand curve. So, now coming to the other points, we can say that in compensated demand curve that is the Hixian demand curve for the change in the income, we have to compensate the consumer, but in Marshallian we are not going to compensate the consumer. And how we have to compensate that if there is a price decrease, then for that price decrease now consumer is having more of that nominal income with that in his pocket. So, we can take some from that that the consumer is now compensated for that incremental change in the real income. And now if the consumer is facing the increase in the prices, so for that increase in the prices we have to compensate the consumer to have that situation that he is having more from the market just to equate that difference. But for that compensation if we are not going to compensate then the Marshallian and the Hixian they are different in that sense that the Marshallian curve it shows the substitution effect and the income effect both. But for the Hixian demand curve it shows only the substitution effect, so therefore its response will be steeper as compared to the Marshallian demand curve. Now coming to this situation, if I explain from this graph, it will be quite easy for us to understand that this situation when the consumer is having good X1 on X axis and the other good on Y axis and the consumer is having this situation when we can say that the consumer is indifferent between the one utility curve and from that utility curve we can assess that if from this budget line consumer is having the situation and he is having equilibrium at point A. But this is the only case when we say that the price of X it falls, so with the decrease in the price of X now consumer has this budget line and as this budget line has the same intercept on X2 but only the price of X1 has changed due to this only the intercept of X1 has changed. Now according to this new budget line consumer shift from this to the higher utility function and approaches to the point that is the B but if we compare that this new budget line is now having the slope that is equal to price of X by price of Y and here that is the price of X prime and this original budget line is having the slope of price of X and price of Y. So keeping in view we can see that this budget line is flatter that is the second part and now if we just move the consumer in that way that he is having a parallel budget line this and this parallel slope budget line is having a slope that is just equal to our second budget line. So now at this budget line consumer is having a tangent point at C means here the slope of utility for line is equal to the slope of second budget line. So movement from point A to C it shows only and only the movement along the same utility function but movement from point A to B it shows the movement that encompasses also the substitution effect and also the from point C to B that is due to the change in the income and when we derive the demand curve mean to these two on the downward axis we see that it is the X1 with respect to the both prices we see that this is the same point of origin that we were having but this compensation point has this demand curve that is the Hixian demand curve and where we have included both substitution and income effect now the response is larger and consumer is having this Marshallian demand curve on the right side. So if this is the case of decrease in income and likewise previously we have shown in the increase in income we can explain that there is a duality between the both. Thank you.