Marginal utility and budget lines
Equalizing Marginal Utility per Dollar Spent Why the marginal utility for dollar spent should be theoritically equal for the last increment of either good purchased
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- 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 we would get for each buck.
- And we saw starting off with our first dollar we got a lot of bang for our buck.
- And this is just another way of saying bang for the buck: marginal utility per price.
- So we got a lot of utility per price starting off with that first chocolate bar, a little less for that next chocolate bar,
- but still more than we would get for a pound of fruit.
- Then more for the next chocolate bar. And then, and only then, did we start buying some fruit, 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 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 plot the marginal utility per price, which is really bang for your buck, on the vertical axis.
- This right over here, on this axis.
- This is the marginal utility per price.
- And let's say it also goes from zero to 100. So that would be 50.
- And the numbers don't actually matter so much here.
- And then this will be dollars spent... dollars spent, or your buck.
- So this is bang for your buck, and this is your buck.
- This is one, two, three, four, five and six. Now we're going to do two 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 or the less you want fruit for that next incremental pound.
- 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's write it this way.
- So this is the marginal utility for A per price of A.
- Now let me get another product right over here.
- So let's say my other product would look 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 that penny?
- And I'm assuming I can buy these in super-small chunks,
- as small as maybe the penny or even fractions of a 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 A and I would get this much bang for my buck.
- Which would be this entire part, this entire part right over here.
- Let me color it in. So first I'll spend it right on A.
- Let me do it in a color that's more likely to be seen, this blue color.
- So I'll spend it on A. In fact, where would I spend my first dollar?
- Well, for the whole first dollar I'm getting a better bang for my buck on A.
- So my first dollar, I'll spend on A. And the total utility I'll 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 obviously give you 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 the 100 marginal utility per price for the entire dollar.
- It's going down the entire time.
- And so your actual total marginal utility is just the area under this.
- And if 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.
- Actually, even after we already spent a dollar 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 till right around there.
- Now something interesting is happening. So we've spent about two dollars.
- 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 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 in on Product B.
- So we can jump right over there and 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 spend 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 a split between Product A and Product B.
- If we spend too much on one and we go down this curve we could've gotten higher utility 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 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 a project -- 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 B was, I don't know, if it was fruit and let's say A was chocolate
- but we could buy it in very, very small increments
- we're saying for that last fraction of a pound of fruit
- you're spending or 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, 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.
- So the general principal: 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 vs. that very last ounce of fruit
- the marginal utility per price for that last increment of one good
- would 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 utililties per 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 - 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 for an incremental of B vs. an incremental of A.
- And at that point 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, at that point you switch and just start buying a bunch of Product B over here.
- Well, that didn't make sense because you were buying Product B
- when you could have 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've gotten higher marginal utility over here.
- This is closer to, I don't know, 75
- while you're only getting 70 right over here.
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