 The Slutsky Identity What is the Slutsky Identity? Up till now, we have Studied that the consumer demand changes in response to three facts Number one its own price So what will be the effect if the price of that good of Which we are going to study if it's going to change What will be the effect of the other goods on The quantity demanded what will be the effect of the income change on the commodity demand when we have to study the effect of the change either of price of x either of price of y or Either the income we have to study only that factor keeping all other factors constant in the position of Thus change in the quantity demanded we can come up with the various solutions So under that various solutions we have studied up till now that the change in the quantity demanded it can be on one path the Direct demand curve and we can solve this directly through the direct demand curve and at the same time this Optimal decision we can tackle this through the indirect utility fund so the total change in the quantity demanded of Change in x is the change due to change in price holding all other factors Constant then we can say that this change is equal to this when there is only Change that we have Included is the change in price when this change in the price It is decomposed or broken into two separate components. We have up till now Achieved that one part will be the substitution effect and the other part that will be the income effect when we add up the income and the Substitution effect we come up with the total effect mean the total change in the quantity Demanded due to change in the price So when we have to answer this type of the question We know that the signs of the income effect and the signs of the substitution effect they matter a lot and The substitution effect it will always be negative means It will be opposite to the change in the price if price will increase Substitution effect will be negative if price will decrease Substitution effect will be positive But the income effect it can be positive if we are dealing with the normal good But the income effect it can be negative if we are dealing with the inferior good. So the total effect in the form of the Normal good it can be positive means both Substitution and the income effect they reinforce each other But in the form of Inferior they both can offset each other So the slutski identity for a normal good if we come up We can say this is a method or the situation when we Calculate the change in the commodity demanded to the price change and Including the substitution and the income effects simultaneously So when we see that if there will be the decrease in the price and We are dealing with the normal good. We see that the price it decreases and Response to this demand it increases and This is the substitution effect and the real income because it Increases due to change in price and due to increase in real income now the demand will increase It means this is the income effect and this to reinforce the total effect will be here and when There will be the increase in the price. It will again will be the substitution effect and Income effect and the total reinforced effect will decrease in the commodity Demanded and here it was increased in the commodity Demanded now when we deal with the commodity the inferior There is only one change for the inferior when price it decreases the consumer Not only Responds in a manner that it decrease its demand. That is a very exceptional case so in this this Substitution effect this and now real income due to the Decrease in the price there will be some other form of the income left with the consumer to spend So due to that really income Here it increases so we are having both Substitution effect and we are having both Income effect, but one form there will be that Substitution effect and the income effect that they will be into the opposite direction to the each other so decrease in the price here It will not have the negative effect of the substitution rather Decrease in price will have the decrease in the demand mean Substitution effect will be positive. So when this will be the shape Here again, not only their sign rather the magnitude of change It will affect because if income effect will be more Then the substitution effect then it will have Total effect means change as compared to their sign and if the magnitude of the income effect will be less Then the sign of substitution effect will prevail and the result and total effect will be different And when we are dealing with the given good then again, we see that now Due to the change in the price decline demand has decreased and Due to the change in the price increase demand has increased so Substitution and the income effect they both move into the opposite direction but Particularly in this case We see that the income effect if it is quite large enough It can offset or it can outweigh the effect of the Substitution effect and the total effect it can be positive. Thus this necessities or it tells that a given good Whenever we are going to study for this it is sure that it should be inferior good But for an inferior good, it is not necessary that it will be a given good so for a given good the Necessary point that it should be an inferior means that it should have our reflection Related to the price response and at the same time we can say that It should not have any type of the substitute or the near substitute Available now if we explain this in the form of our diagram We can see that mostly the directions of the chain They are similar but with certain exceptions that it retains to the level of the magnitude If the diagram on this left-side panel we see that the consumer is having this original budget line and With this original budget line now we see That if price of commodity x1 it decreases So when it decreases we can have income divided by a price of x it means now Due to the smaller level of denominator we will have more of the x1 so Consumer it shifts from this budget line to this budget line and When it moves to this budget line it has different slope when equal to this budget line slope we draw a parallel Hypothetical budget line on the previous indifference curve and on the same indifference curve Having the Hypothetical or the expected budget line of the slope now the consumer shift from this point a To this point B in the first stage That is the pivot movement that will be reflected through substitution effect But after this substitution effect now the consumer Has shifted to this point B in a manner That now consumer has shown not only increase in the income rather Consumer has shown a total decrease in the backward form here and Likewise, we see that The similar nature of the response is shown in this panel where it was the original budget line and Consumer shift to this budget line due to decline in the price of x1 and With the hypothetical budget line consumer shifted from a to B and from a to B Shift was due to substitution effect and from B to C the shift was due to Income effect, but we see that in this panel from B to C there is Similar nature of the change But the magnitude of this change is smaller in the form of only inferior goods But in the form of the given case this magnitude is larger as Compared to the inferior it means for the given goods It is necessary that the given it should be inferior also But for the inferior commodity it is not compulsory for the commodity to be given