Utility maximization: equalizing marginal utility per dollar

In the last video, we thought about how we would allocate our $5 between chocolate bars and fruit. And the way we did it, and 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 got 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 for price starting off for that first chocolate bar. A little less for the next chocolate bar, but still more than we would get for a pound of fruit. Then more for the next chocolate bar, and only then did we start buying some fruit, buying some pounds of fruit. What do I do in this video is generalize it. I want to think about maybe a more continuous case where we can buy very, very small increments of each of the products. It doesn't have to be in chunks, like chocolate bars. And what I'm going to do is I'm 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 spent, so your buck. So this is bang for your buck and then this is your buck. So this is 1, 2, 3, 4, 5 and 6. Now we're going to do arbitrary products. So let's say one product looks something like this. And once again, you have diminishing utility as you get more and more of that product. In the case of fruit, the more pounds of fruit you get the more tired you get of fruit. The less fruit you need for that, or the less you want fruit for that next incremental pound. So let's-- but it could be anything. This is true of most things. So this is product A, could be a service as well. So product A, let me write it this way. So this is the marginal utility for A per 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 my marginal utility for product B per price of B. So it's really saying bang for the buck. So just to start off-- and 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 a penny. And I'm assuming I can buy these in super small chunks, as small as maybe the penny or even maybe fractions of penny. So if I just had a penny, and I had to 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 A and I would get this much bang for my buck, 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 in 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 with 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 100 marginal utility per price for the entire dollar. It's going down the entire time. And so your actual total marginal utility is actually just the area under this. And when you take calculus you'll get a better appreciation for that. But let's just think about, once again, where our dollar is going to be spent. So actually even if we've spent already $1, our next penny we would 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've spend our first $2 all on product A because we're getting more bang for 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 penny? Well, we could spend it on product A again. But look, we can get about the same marginal utility spending it on product B. 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 of product B, or a little bit more of 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 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 this one. If we spend 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 spent 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 $2-- there's an interesting thing. The marginal utility for B, or the marginal utility for price for B that I spent on that last little increment is going to be the same as the 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 could buy them in very, very small increments-- we're saying for that last fraction of a pound of fruit 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 utility for price for that last increment of one good will be the same as 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 utility for price of the two goods are the same. And not even that one is better than the other. This is just saying as you spend money, and 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 for price is the same for an incremental of B versus an incremental of A. And at that point you're juts 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 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 70 right over here.