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### Course: Microeconomics>Unit 5

Lesson 1: Introduction to consumer theory: total utility and marginal utility

# Utility maximization: equalizing marginal utility per dollar

How should you allocate money between two different products or services in order to maximize utility, or "bang for your buck?" Marginal utility refers to the utility gained from an incremental purchase, in order to make rational decisions. In many cases, the goal is to reach a point where the marginal utility per price of the two products is equal. Created by Sal Khan.

## Want to join the conversation?

• Does the point where the graphs crosses have any meaning in this case?
(44 votes)
• RIP, you are wrong. Even after the intercept, you will keep switching between A and B, since, for example, B at \$6 is lower in MU/P than A at \$3.10.
(21 votes)
• what is equi-marginal principle?
(5 votes)
• the equi-marginal principle is what Sal is explaining here. the fact that MUa/Pa = MUb/Pb. If one good has a better marginal utility, then you would buy more of that good, decreasing the marginal utility of one more unit of that good. However, the best situation would be where you get the same "bang for your buck" from both goods. This point where the two sides of the equation are the same. the point where the marginal utility per price is equal (equi-marginal principle).
(14 votes)
• What is marginal utility?
(6 votes)
• Margin means edge or the next one. Marginal utility is the utility you receive from the next one or "at the margin." In economics it is often assumed that consumers maximize their utility at the margin or get the best deal for the next dollar spent.
Maximizing utility at the margin isn't necessarily simple. Do you maximize the utility of each item you buy at the store or do you get the most for a fixed amount you intend to spend so the margin is the shopping trip not each item.
(6 votes)
• Is there a way that by purchasing one product or another, that the other product's marginal utility would be altered? I could see complimentary items being a common positive example where by purchasing a car your marginal utility for gas increases, but is there a negative example? Where by purchasing one item, another is worth less to you? Because in that case, wouldn't the rule that the two curves equalize at some point be negated?
(4 votes)
• Sal probably implicitly uses "Ceteris paribus" (all things being equal) here for simplicity. It's easy to imagine lots of goods with dependent marginal utility values. In that case, graphs would move in a more "dynamic" way. The more movies you rent online, the less demand you have for movie theater tickets. The more water filters you get, the less need there is for purchasing water bottles. On the other hand, we can also consider human psychology affecting things in perhaps unpredictable ways. For every 10 movies watched at home (e.g. renting DVD's), your desire to watch at the theater slowly increases; after all, home experience can't compare to the theater one. Things that seem interchangeable could have an unexpected effect on each other.
(5 votes)
• i'm a bit confused. according to the graph, after about \$4 you will no longer switch between A and B and just start purchasing B right?

at that point, it seems that the marginal utility of A stops becoming equal to the marginal utility of B (by drawing a straight line through the two lines).

am i wrong?
(3 votes)
• No you keep switching all the time. Even after the crossing point. Thats the whole idea. Keep in mind the things you have to compare here:
You dont compare the utility for each product at 4 \$. You rather compare the utility with the last of the 4 dollars spent at total. so maybe you spend 2 on product A and 1 on Product B. Now you have one more dollar. you can spend that dollar at A (that would be the third "A dollar" or on B (that would be the second "B dollar". So you compare the Y values of B (at X=3) with the one of A (at X=1). You dont compare the Y values at 4...
(2 votes)
• so our utility is maximized in the equalibrium? where the two marginal utility per dollar spent- curves cross each other?

or is it wrong, like: we should allocate our money, in other words spend our money where the mu/price for the two goods is bigger? and we should stop doing this where the equalibrium starts?

i hope you understand my question, my english is not my first language.
(2 votes)
• shouldn't the Y axis just be MU not MU/P?
(2 votes)
• Simple but interesting question. Sal is graphing `MU/P`, because he's not expressing what the MU is directly. The graph is for the amount of `MU` for any incremental spending, at any given amount that you already spent (`P`). He did mention calculus, but I think he tried to get through the video without explaining that by integrating `MU/P` for change in P, you get an area that's expressed in `MU`. However, despite all that, I think Sal used the term "marginal utility" incorrectly. At he describes the area under the curve as "total marginal utility", but it should just be "total utility". (I think he did it a bit strangely here in order to emphasize that the marginal utility of each curve was expressed per price, not per pound or per unit.)
(1 vote)
• I learned from a textbook that the formula derived at the end is the 'utility maximizing rule'. Could somebody please explain this?
(1 vote)
• Is there a way to find how to maximize total utility without drawing a graph?
(1 vote)
• Yes, there are several ways to maximize total utility without drawing a graph. Here are some common methods:

1. Marginal Analysis: Calculate the marginal utility of each additional unit of a good obtained and compare it with its price. Allocate your resources in a way that the marginal utility per dollar spent is equal for all goods. This ensures that you are getting the maximum level of satisfaction from your expenditure.

2. Budget Constraint: Consider your budget constraint and the prices of different goods. Allocate your resources in a way that maximizes your overall utility while staying within your budget. Look for combinations of goods that give you the highest level of satisfaction without exceeding your financial constraints.

3. Consumer Preferences: Determine your preferences for different goods and services. Identify the most preferred combinations that maximize your utility. For example, if you have a preference for good A over good B, allocate your resources in a way that allows you to consume more of good A relative to good B.

4. Use Mathematical Optimization: Apply mathematical optimization techniques, such as linear programming or calculus, to solve utility maximization problems. These methods involve setting up equations or inequalities that represent your preferences and constraints and then finding the optimal solution mathematically.

Each of these approaches provides a way to determine how to maximize total utility based on your preferences, constraints, and available information, without explicitly drawing a graph.
(1 vote)
• Wouldn't the 5th dpollar be spent on the next unit of fruit at 50 utils per price? Why was the last dollar spent on the same unit at 60 utils?
(1 vote)
• One pound of fruit was 120 utils total, for \$2. So, actually, Sal spent the last two dollars on the same pound of fruit to get 120 utils.
(1 vote)

## Video transcript

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.