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Utility maximization: equalizing marginal utility per dollar

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Video transcript

in the last video we thought about how we would allocate our five dollars between chocolate bars and fruit and the way we did it was very rational we thought about how much bang would we get for each buck and we saw look starting off our first dollar we had a lot of bang for our buck and this is really just another way of saying bang for the buck marginal utility per price so we got a lot of utility propriety starting off for that first chocolate bar a little less for that next chocolate bar but still more than we would get per for a pound of fruit then we get it we then more for the next chocolate bar and only then and only then did we start buying some fruit buying some pounds of fruit what I want to do in this video is generalize it I want to think about maybe a more continuous case where we can buy very very very small increments of each of the products it doesn't have to be in chunks like chocolate bars and then and what I'm going to do is I'm going to going to plot the marginal utility per price which is really bang for your buck on the vertical axis so this right over here on this axis let's say this is the marginal utility per price and let's say it also goes from 0 to 100 so that would be 50 and the numbers actually don't matter so much here and then this will be dollar spent so dollars dollars spent so your buck so this is bang for your buck and that this is your buck so this is one two three four five and six now we're going to to arbitrary products so let's say one product looks something like this and then once again you have diminishing utility as you get more and more of that product in the case of through the more pounds of fruit you get the more tired you get a fruit the less you the less fruit you need for that or or the less you want fruit for that next incremental pound so let's but it does it could be anything this is true of most things so this is product a it could be a service as well so product a let me write it this way so this is the marginal utility for a / price / price of a and let me get another product right over here so let's say my other product looks something like this so this is this is my marginal utility for product B the per price of B so it's really saying bang for the buck so just to start off just to start off and I won't even I won't even constrain how much money we have I just want to think about how we would spend that money so if I were to spend if I had a penny where would I spend that penny and I'm assuming I can buy these in super small chunks the smallest maybe the the penny or even maybe fractions of penny so if I just had a penny and I think about where am I getting the best bang for my buck for that penny I'm clearly getting it with product a so I would spend that penny on product day and I would get this much bang for my buck which would which would be this entire part which would be this entire part right over here let me color it in so my first I'll spend it right on a let me do it a color that's more likely to be seen so I'll do it in this blue color so I'll spend it on a my first in fact where would I spend my first dollar well the whole first dollar I'm getting a better bang for my buck on a so my first dollar I will spend on a and the total utility I will get is actually going to be the area under this curve it's going to be this whole area it's going to be dollars times marginal utility price that would give you obviously the area of this rectangle right over here the reason why it wouldn't be the area of this larger rectangle it would just be the area under the curve is you're not getting you're not getting this a hundred marginal utility per price for the entire dollar it's going down the entire time and so that your actual total marginal utility is actually just the area under this and if you when you take calculus you'll get better appreciation for that but let's just think about once again where our dollar is going to be spent so our actually even our after even we spent already $1 our next penny we'd still want to spend on product a because we're still getting more bang for the buck we're still getting more bang for the buck all the way until right around there now something interesting is happening so we've spent about $2 we spent our first two dollars all on product a because we're getting more bang for our buck even though that bang was diminishing every penny or even every fraction of a penny that we spent but now where will we spend our next our next penny well we could spend it on product a again but look we can get about the same marginal utility spending it on be so we could jump right over there spend it on product B now where could we spend our next dollar well we get about the same marginal utility whether we spend it on a little bit more product B or a little bit more product a so we could do either if we spent a little bit too much on product a then we could have gotten more marginal utility spending on product B so what we would do is once we've gotten to this threshold right about right about here we actually are going to spend every incremental fraction of a penny we're actually going to want to split between product a and product B if we spend too much on one and we go down this curve we could have gotten higher utility spending on on on this one if we spent too much on this one we could get higher utility spending on this one right over here so there's a very interesting phenomenon here assuming that we eventually spend enough that we buy some of both obviously we started just buying product a because it had higher utility at least for those first few dollars but assuming that we end up buying some mix of the two which we do end up spending if we spend more than two dollars there's an interesting thing the marginal utility for project the marginal utility for for B or the marginal utility for price for B that I spend on that last little increment is going to be the same as a marginal utility per price for that last increment of a so if this was if B was I don't know if it was fruit and let's say a was chocolate but we can buy him in very very small increments we're saying for that last fraction of a pound of of fruit you're spending thus it you're getting the same marginal utility per price as you're getting for that last fraction of a bar or fraction of a pound of chocolate so there's a general principle over here and it really just comes from this very straightforward thing that as soon as you can get better marginal utility on the other one you start spending there but then they start to look equal and you would keep dividing your money between the two and so the general principle if you're allocating money between two goods for that last increment not across the board just that last increment that's why the word marginal is so important for that last ounce of chocolate versus that very last ounce of fruit the marginal the marginal utility per price for that last increment of one good will be the same as the marginal the marginal utility per price of the second good now I really want to emphasize what this is saying this is not saying that the marginal utilities for price of the two goods are the same and not even that their one is better than the other this is just saying as you spend money let's say you spend enough money to buy both at some point you're going to get to a threshold where you're neutral between the two where the marginal utility per price is the same at for eat for an incremental of B versus an incremental of a and at that point you're just going to keep switching you're just going to keep switching between the two products because obviously if you focus too much on this right over here let's say you focus let's say at that point you switch and you just start buying a bunch of you just start buying a bunch of product B right over here well that didn't make sense because you were buying product B when you could have actually gotten higher marginal utility buying some of product a and that's the same reason why you didn't just keep going down a because you could have gotten higher marginal utility over here this is closer to I don't know 75 while you're only getting seventy right over here
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