Visualizing marginal utility MU and total utility TU functions

What we're going to do is think about the graphs of marginal utility and total utility curves. And so right over here I have a table showing me the marginal utility I get from getting tennis balls. And so it says look, if I have no tennis balls and I'm not able to play tennis, so I'm pretty happy when I get that first ball. It gives me a marginal utility of 100.

Now you might say a hundred what? Sal and I would say to you, well that's the thing about marginal utility. We're using fairly abstract units here. We're not speaking in terms of dollar or opportunity cost. We're just have this abstract unit we could call it utility units if we want, but what really matters is the relation between these values.

So for example, that second ball, it's nice. Now I can lose that first ball or I don't have to chase that first ball around as frequently, but it's not as—I don't get as much incremental utility as I got from that first ball. That first ball allowed me to play tennis. Now the second ball, I was already playing tennis; now my tennis is just going to be a little bit more pleasant.

So the marginal utility, one way to think about it, this is 80 and this is 100. The marginal utility of that second ball is eighty percent of what that first ball is. Now I promise that I would graph these so let's get a graph out here.

So there we go, and let's start plotting these and see what it looks like. Well, that first ball gave me a utility of a hundred. The second ball gives me a utility of 80. Third ball—you see the trend—this is a downward sloping. If this was a line, these are discrete values here, but if I cared about half of a tennis ball or three-fourths of a tennis ball, then I could connect the dots here as well. But you can see that it is a downward sloping line.

If you were to connect the dots, so three balls utility of 60. If I have four balls, it's a utility of 40. Five balls utility of 20. And once again, when I get that fifth ball, yeah, it's nice, but now my pockets are getting pretty full and I was all, it's hard to play tennis with those full pockets.

By the time I get that sixth ball, I get no marginal utility from that. I don't have a place to store it; I don't really—I'm not really into it. And then that seventh ball, I actually view it as a negative. It's one extra ball to worry about, no place to store it; it's taking up space in my life.

And so if you were to connect the dots, we're talking about a discrete case, but oftentimes you could think about something that's a little bit more continuous or the quantity is more granular. If you said pounds of chocolate or something like that, then you could imagine, in general, for a marginal utility curve to be able to connect these dots.

And you see in this case it is downward sloping. So this is the marginal utility, the marginal utility curve. Notice that it slopes down, slopes down. And this you're generally going to see for any marginal utility curve because the incremental benefit of that next amount, that next unit, is seldom as good as the benefit of getting it before.

You get tired of the thing, you start running out of space, you've already consumed more than you need of it. And this is consistent with what we know from the law of demand. In the law of demand, every incremental amount of quantity people like—well, I already have some. And this might not be—the law of demand is not talking about just an individual; it's talking about a market.

But the market as a whole is made up of individuals. And if each individual is saying, hey, you know, that first unit really matters to me, but the next unit, it's nice but not as good. So I'd pay less for it. And then the unit after that, it's nice so I might pay less for it. And so if you aggregate all those individuals, that's where that law of demand comes from.

These are consistent ideas, and that's why the demand curve, you also, it is sloping down. Price is usually on this axis instead of utility, and you could imagine price as being a proxy for utility, and quantity, of course, is on this axis right over here.

So this is quantity of balls. So that's our marginal utility curve. What about total utility? So let me have a table here that shows total utility.

And total utility for marginal utility is pretty straightforward. All you do is say, okay, well that first ball, when I have one ball, my total utility is the same as my marginal utility. And so you're going to have that same starting place at when your consumption is just beginning.

But then your total utility from two balls, well I had a hundred utility units from the first ball, and then I get 80 more from that second ball, so it's going to be 180. So for two balls, my total utility is 180. All I'm doing is I just added that to that.

Now for three balls, I add that to that to that. I take 180 and I add the 60 extra utility units I get for that third ball, and now I'm at 240. So that third ball gives me, gets me to 240 right over there.

Now the fourth ball, once again, I'm just going to take 240 and add the incremental utility of that fourth ball, the marginal utility of that fourth ball; that gets me to 280. So that gets me to 280 which is right over—let's see—20, 30, 20, 40, 60, 80. So this is right over there.

Now that fifth ball, I'm just going to take the 280 and then the marginal utility of the fifth ball, add 20, gets me to 300. Fifth ball gets me to 300, which would be right around there.

Now the sixth ball gave me no incremental, no marginal utility, so my total utility when I have six balls stays the same. I'm indifferent as to whether I have five or six balls. So the sixth ball is now flat right over here.

And now the seventh ball, I'm tired of these tennis balls; I'm being overwhelmed by them. I'm finding it stressful, and so it actually has a negative marginal utility. And so my total utility, if someone forced me to have seven balls, my total utility would now go down by 20.

And so my total utility now would be 280 right over here. And you could see the marginal utilities here. If you just say, look, this is plus 80. This is plus 60. This is plus 40. This is plus 20. This is plus zero, and then this is minus 20. And so you see the numbers right over right over there.

Now this tennis ball example, this would be a discrete case. You wouldn't have half a tennis ball or pi tennis balls or something like that. But if we want to speak in general terms, you could think about connecting the dots if you had a more continuous example.

And your total utility curve might look something like this. Now what's interesting is right when you're beginning consumption, you're starting at the same place. Well, that makes sense; your first unit, yet marginal utility, that's going to be your total utility.

And this is upward sloping as long as you're getting some positive marginal utility from each increment. So as long as my marginal utilities were positive, well, this graph is going to be increasing. But notice, because the marginal utilities are getting—are decreasing right over here, the rate of increase for total utility is decreasing.

The slope here is decreasing. You can view the marginal utility as the slope of the total utility curve. And then notice the total utility curve has a maximum value. It's starting to hit a maximum value right over there when the marginal utility curve is hitting zero.

Because beyond that point, where at least in this example we had negative marginal utility. And so when you add that seventh unit, well, that's going to make your total utility curve go down. And so you're going to have a negative slope in this particular example.