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# Consumption function with income dependent taxes

## Video transcript

in the last video where we generalized the linear consumption function I said I said that the tax the total amount of taxes the aggregate taxes are constant all of these two were constants right here so you can kind of merge them into a constant that ended up being our our independent variable intercept right over here but YouTube user nil soar 1337 asks a very interesting and good question well aren't taxes in some way a function of aggregate income in most in most modern economies people pay a percentage of their income or in general tax the tax base grows as aggregate income or as GDP grows so is it appropriate to make this constant so the simple answer is it depends on how carefully you want to model it in some cases you might just say well let's just assume that this is a bulk tax we're just trying to understand one aspect of it and you will see that in some economics courses or some economics textbooks the other way is you could actually model it a little bit more realistic you could say hey taxes really are a function of aggregate income we could say that T really is going to be equal to some tax rate I'll write that as a lowercase T times aggregate times aggregate income and in the place like the US this might be close to the 30 percent of aggregate income or 20 percent whatever it might be of aggregate income is what is going to go for taxes and if you do it this way and you substitute back into this you can actually get an expression of for consumption in terms of aggregate income that takes into factor takes into factor that takes into consideration the idea that taxes are function of aggregate income and just to do that algebraically we can rewrite this expression up here so you have consumption aggregate consumption is equal to is equal to my marginal propensity to consume times aggregate income times aggregate income plus plus autonomous consumption the amount that would be consumed no matter what - minus the marginal propensity to consume shows up again and then instead of writing T instead of writing T right over here I'm going to write I'm going to write lowercase T times y so the tax rate times aggregate so times the tax rate times aggregate times aggregate income so I just took this instead of writing uppercase T I look I wrote lowercase T times aggregate income and they should be the same thing but now we've expressed T as a function of aggregate income and now we can merge both of these these are something times aggregate income so we can combine those two terms so this one and this one right over here if we factor out if we factor out a common factor of C one times y we get let me write it this way actually let me just combine them first so that the algebra doesn't confuse you so we get C is equal to is equal to c1 times y so marginal propensity to consume times aggregate income now I'm going to write this one minus the marginal propensity to consume times I'll switch the order here well let me not switch the order times the tax rate not just the aggregate total tax value but the actual tax rate times times aggregate income times aggregate income so that's those two terms there and then we're just left with the autonomous consumption so plus the autonomous consumption and now over here we have a common factor of we can factor out a c1 and and the y or essentially the marginal propensity to consume and the aggregate income and so this is just algebraic manipulation right over here we get aggregate consumption is equal to is equal to let's see we could write this as C 1 times 1 minus T 1 minus T times y times y you can multiply this out to verify if you multiply it all out then the first term is C 1 so C 1 times 1 times y is this right over here and C 1 times negative T times y is this term right over here and then you're left with then you are left with your autonomous consumption and this actually makes a lot of sense because when you write it like this when you write it like this you can look at this term right over here what is this term right over here well 1 minus the tax rate times aggregate income if the tax rate is 30% then one minus 30% is going to be 70 percent 70 percent times aggregate income that's essentially what people get in their pockets this is this whole term right over here is essentially a disposable income disposable disposable income right over here so we could actually if we wanted to write this as some other variable we can just put that variable right over there and say it's disposable income and then actually becomes a very simple thing to graph so we could graph this two different ways if we wanted to write it a function of aggregate income of aggregate income we would we would graph it we would graph it like this now when we express it this way taxes as a function of aggregate income now our vertical intercept so this is aggregate consumption our vertical intercept is this term right over here so that is C naught and our slope is all of this business and so the slope of our line the slope slope is going to be C 1 times 1 minus T and this right over here the independent variable is aggregate income another option we could set some other variable to say well we could say disposable income let me call it Y Y disposable is equal to 1 minus T times y and then we could write this so it's essentially equal to this business right over there so then we can rewrite the consumption function as aggregate consumption is equal to the marginal propensity to consume times disposable income times disposable income plus some level of autonomous consumption plus plus some level of autonomous consumption and this actually takes us back to the basics this takes us back to our very original situation here where we had some autonomous consumption plus our marginal propensity to consume times disposable income and if we wanted to plot it this way as a function of disposable income not aggregate income then it would look like this so this is consumption and now this is an aggregate income this is disposable income disposable income which is the same thing as 1 minus the tax rate times aggregate income now still our vertical intercept depth is see not and our lines slope is the marginal propensity to consume so this is see one just like that so all of these are completely valid consumption functions and I thank nil sorr 1337 for bringing up a topic that actually was a cause of confusion for me because it really does depend because I thought the way he or she originally thought about the problem is that well taxes are a function and a lot of econ books tend to treat this as a constant and that is actually just an assumption they make to often simplify the calculations but if they don't want to make that assumption you can still show that though you can still show that it is a linear function of that consumption is still a linear function of aggregate income