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# Generalized linear consumption function

## Video transcript

in the last video we began our exploration of what a consumption function is that's a fairly straightforward idea it's a function that describes how aggregate income can drive aggregate consumption and we started with a fairly simple model of this a fairly simple consumption function it was a linear one you had some base level of consumption regardless of aggregate income and then you had some level of consumption that was essentially induced by having some disposable income and when we plotted this linear model we got a line we got a line right over here and I pointed out in the last video this does not have to be the only way that a consumption function can be described you might use some fancier mathematical tools maybe you can construct a consumption function you have an argument you'd argue that the marginal propensity to consume is is higher at lower levels of disposable income and then it it kind of tapers out as disposable income as aggregate disposable income goes up and so you might think that maybe you should have a fancier consumption function that when you graph it would look like this and then you would have to use things fancier than just what we used right over here what I want to do in this video is focus more on a linear model and the reason why I'm we're going to focus on a linear model is because one it's simpler it'll be easier to manipulate and it's also the model that tends to be used right when people are starting to digest things like consumption functions and building on them to learn about things like and we'll do this in a few videos the Keynesian cross but what I'm going to do is I'm going to do two things I'm going to generalize this linear consumption function and I'm going to make it a function not just a disposable income but not just a aggregate disposable income which is what we did in the last video but as a function of income of aggregate income and then we will plot that generalized one based on the variables but it's really going to be the same thing we're just not going to use these numbers we're going to use variables in their place so let's give ourselves a linear consumption function so we could say that aggregate consumption aggregate consumption we're going to have some base level of consumption no matter what even if people have no aggregate income they need to survive they need food on the table and maybe they'll have to dig in savings somehow to do it so some base level of consumption I'll call that lowercase e Sub Zero or lowercase e with a subscript of zero right over there that's the base level of aggregate consumption or sometimes referred to as autonomous consumption this is autonomous autonomous consumption because people will do it on their own or in aggregate they will do it on their own even if they have no aggregate income and then we will have the part that is due directly due to having some aggregate income and we call that the induced consumption because you could view it as being induced by having some aggregate income so above and beyond what the base level of consumption people are going to consume some fraction of their disposable income so we'll say disposable disposable disposable income and they're not going to consume all of their disposable income they might save some of it so they're going to consume the fraction that's essentially their marginal propensity to consume so this right over here I'll do that in I'll do that in this orange color marginal propensity to consume so that is marginal propensity to consume and hopefully this makes intuitive sense this says look if this was a hundred people are going to consume 100 no matter what a hundred billion whatever your unit of currency is now their marginal propensity to consume is let's say it is a third and you have now above and beyond this people have disposable income of let's say 900 this is saying that they're going to they want to consume a third of that of that disposable income they're getting that is if you give them 900 of extra disposable income their propensity to consume that marginal in that that incremental income they're going to consume a third of it and so this would be a third so it would be nine hundred so let me give an example so if you had a situation so you could have a situation where C naught is equal to what did I say I say it was like 100 and if you have disposable income disposable income is equal to 900 and C 1 is equal to one third or we could say 0.333 repeating forever C 1 is a third then this makes sense on their own people would consume this much but now they have this disposable income there are marginal propensity to consume if you give them 900 extra of income they're going to consume a third of that so then you're going to have your consumption is going to be equal to for this case right over here your consumption is going to be 100 times or plus 1/3 plus 1/3 times 900 times 900 and so your consumption in this situation your induced consumption 1/3 times 900 would be 300 and maybe it's in billions of dollars 300 billion dollars and then your autonomous consumption would be 100 and they would add up to they would add up to 400 so once again this is autonomous and this is induced autonomous this right over here is induced induced consumption consumption now I did write it in general terms I'm using variables here instead of instead of or constants really instead of using instead of using the numbers that we saw in the last example but I also said that I would express aggregate consumption as a function not just a disposable income but of aggregate income not just of the aggregate disposable income but aggregate income and the relationship is fairly simple between disposable income and overall income we saw over here in aggregate you have income but the government in most modern economies takes some fraction of that out for taxes and what's left over is disposable income and so and just reminder income in aggregate aggregate income is the same thing as aggregate expenditures which is the same thing as aggregate output this right over here is GDP so this right over here is let me do this in a color I've used almost all my colors this is equal to GDP so disposable income is essentially GDP or you could say aggregate income minus taxes - - I'll do the taxes in a different color - taxes - taxes so we can express disposable income as aggregate income this right over here is the same thing as aggregate income minus taxes - taxes so we can rewrite our whole thing over again aggregate consumption is equal to autonomous consumption plus the marginal propensity to consume times times aggregate income which is the same thing as GDP times aggregate income minus taxes - taxes and so we fully generalized our consumption function and now we've written it as a function of aggregate income not just aggregate disposable income and to make you comfortable that this is still aligned if we write it if we were to plot it as a function of aggregate income instead of disposable income let me manipulate this thing a little bit so we could distribute C 1 which is our marginal propensity to consume and we get aggregate consumption is equal to autonomous consumption and then we're going to distribute this plus C so we're going to multiply it times both of these terms plus our marginal propensity to consume times aggregate income times aggregate income and then minus our marginal propensity to consume times our taxes times our taxes and since we're going to where we want it as a function of aggregate income everything else here is really a constant we're assuming that those aren't going to change those are constant variables and so what we could do is we can rewrite this in a form that you're probably familiar with back in algebra class you probably remember line you can write it in the form y is equal to MX plus B where if where X is the independent variable Y is the dependent variable if you were to plot this one the horizontal axis is your x-axis your x axis vertical axis is your Y axis that this right over here would have a y-intercept or your vertical axis intercept of B right over there and then it would be a line with slope M so if you were to take your rise divided by your run or how much you move up when you move to the right a certain amount that gives you that gives you your M slope is equal to M so the same analogies here we can rewrite this in that form where our dependent variable is no longer Y our dependent variable is aggregate consumption and our independent variable is not X it is aggregate income so let's write it in that form so we can write it as dependent variable C which we'll plot on the vertical axis is equal to it's going to be equal to the marginal propensity to consume times aggregate income times aggregate income let's do that purple color times aggregate income plus autonomous consumption plus autonomous consumption minus marginal propensity to consume times taxes so minus marginal propensity to consume times taxes now it looks all complicated but you just have to realize that this part right over here this is all constant this is all constant and it is analogous to the B it's analogous to the B if you were to write things in kind of traditional slope-intercept form right over here so when we plot the line if you have no if you have no aggregate income this is what your consumption is going to be so let me draw that so once again our independent our dependent variable I should say is aggregate consumption and our independent variable in this is no longer a disposable income like we did in the last video it is now aggregate income it is now aggregate income it is aggregate income so there's no aggregate income this is the independent variable right over here if there's no aggregate income then your consumption is just going to be this value right over here so your consumption is just going to be that value right over there which is C naught minus c1 times t and then as as you have larger as you have larger values of aggregate income c1 that fraction of it is what's going to be contribute to the induced consumption and so what you essentially have is this is the slope of our line this right over here is our slope just to kind of draw the analogy if you would say you know y is equal to MX plus B or actually maybe I'll write it like this if you were to write C C is equal to M and I don't want to confuse you if this M and B seem like completely four and this is where it comes from kind of a traditional algebra grounding and slope and y-intercept so if I were to say is C is equal to M y plus B plus B this is the slope this is our vertical or our dependent variable intercept right over here that's where we intercept the dependent variable axis and this is our slope it's our marginal propensity to consume so our line will look something like this where the slope slope is equal to the marginal propensity to consume which is equal to c1 if people all of a sudden are more likely to spend a larger fraction of their income then the marginal propensity of consume are gional propensity to consume would be higher and our slope would be higher we would have a line that looks like that and we always assume that the marginal propensity to consume it will be less than 1 so we'll never have a slope of 1 but we'll also never have a negative slope because we assume that this is positive and if people are more likely to save then consume when they have extra income then this line might look something like that might have a lower slope