[00:11] Um
[00:12] Uh today we're going to start talking
[00:14] about what's underneath the demand
[00:16] curve.
[00:17] So, basically what we did last time and
[00:19] what you did in section on Friday is
[00:21] talk about sort of the workhorse model
[00:24] uh of economics, which is the supply and
[00:25] demand model. And we always start the
[00:27] class with that cuz that's the the model
[00:30] in the course. But, I think as any good
[00:32] sort of scientist and inquisitive minds,
[00:34] you're probably immediately asking,
[00:35] "Well, where do these supply and demand
[00:37] curves come from? They don't just come
[00:38] out of thin air. Uh
[00:41] how do we think about them? Where do
[00:42] they come from?" And that's what we'll
[00:43] spend the base of the first half of the
[00:45] course going through.
[00:46] And so, we're going to start today with
[00:48] the demand curve.
[00:49] And the demand curve is going to come
[00:51] from how consumers make choices.
[00:55] Okay? And that will help us derive the
[00:57] demand curve. Then we'll turn next to
[00:58] supply curve, which will come from how
[01:01] firms make uh production decisions.
[01:04] But, let's start with the demand curve.
[01:06] And we're going to start by talking
[01:07] today about people's preferences
[01:09] and then the utility functions.
[01:11] Okay?
[01:12] So, our model of consumer
[01:14] decision-making
[01:16] is going to be a model of utility
[01:18] maximization.
[01:21] That's going to be our fundamental
[01:22] Remember, this course is all about
[01:23] constrained maximization. Our model
[01:25] today is going to be a model of utility
[01:26] maximization.
[01:28] And this model is going to have two
[01:29] components.
[01:30] There's going to be consumer
[01:31] preferences,
[01:34] which is what people want, and there's
[01:37] going to be a budget constraint, which
[01:39] is what they can afford.
[01:43] And we're going to put these two things
[01:45] together. We're going to maximize
[01:47] people's happiness or their choice or
[01:49] their their happiness given their
[01:50] preferences subject to the budget
[01:53] constraint they face. And that's going
[01:55] to be the constrained maximization
[01:56] exercise that actually through the magic
[01:58] of economics is going to yield the
[02:00] demand curve. And it's going to yield a
[02:01] very sensible demand curve that you'll
[02:02] understand intuitively.
[02:05] Now, so what we're going to do is do
[02:07] this in three steps.
[02:09] Step one over the next two lectures.
[02:11] Step one is we'll talk about
[02:12] preferences.
[02:13] How do we model people's tastes?
[02:16] We'll do that today.
[02:18] Step two is we'll talk about how we
[02:19] translate this to utility function. How
[02:22] we mathematically represent people's
[02:24] preferences in the utility function.
[02:26] We'll do that today as well.
[02:28] And then next time, we'll talk about the
[02:30] budget constraints that people face.
[02:33] So, today we're going to talk about the
[02:34] maximand. Next time we'll talk about the
[02:36] budget constraint.
[02:38] That means today's lecture is quite fun.
[02:40] Today's lecture is about unconstrained
[02:42] choice. We're not going to worry at all
[02:44] about what you can afford, what anything
[02:45] costs.
[02:47] We'll worry about what things cost.
[02:48] We're not going to worry about what you
[02:48] can afford. Okay? Today's the lecture
[02:51] where you won the lottery.
[02:52] Okay? You won the lottery, money is no
[02:55] object. How do you think about what you
[02:57] want?
[02:58] Okay? Next time we'll say, "Well, you
[02:59] didn't win the lottery." In fact, as
[03:00] we'll learn later in the semester, no
[03:02] one wins the lottery. Uh that's an
[03:03] incredibly bad deal. Um uh but basically
[03:07] next time we'll impose the budget
[03:08] constraints. But for today, we're just
[03:10] going to ignore that and talk about what
[03:13] do you want?
[03:14] Okay? And to start this, we're going to
[03:17] start with the series of preference
[03:20] assumptions.
[03:25] A series Remember, as I talked about
[03:27] last time, models rely on simplifying
[03:30] assumptions. Otherwise, we could never
[03:31] write down a model. It would go on
[03:32] forever.
[03:33] Okay? And the key question is, are those
[03:36] simplifying assumptions sensible? Uh do
[03:39] they do violence to reality in a way
[03:41] which makes you not believe the model?
[03:43] Or do are they roughly consistent with
[03:44] reality in a way that allows you to go
[03:46] on with the model?
[03:47] Okay? And we're going to impose three
[03:49] preference assumptions, which I hope
[03:52] will not violate your sense of
[03:53] reasonableness.
[03:54] The first is completeness.
[04:00] What I mean by that is you have
[04:03] preferences over any set of goods you
[04:06] might choose from.
[04:08] You might be indifferent. You might say,
[04:10] "I like A as much as B." But, you can't
[04:12] say, "I don't care." or "I don't know."
[04:15] You say, "I don't care." That's
[04:16] indifference. You can't say, "I don't
[04:17] know." You can't literally say, "I don't
[04:19] know how I feel about this." Um you
[04:21] might uh say you're indifferent to
[04:23] things, but you won't say, "I don't know
[04:25] uh how I feel about something." That's
[04:26] completeness.
[04:28] Okay? The second is the assumption we've
[04:30] all become familiar with since
[04:31] kindergarten math, which is
[04:33] transitivity.
[04:38] If you prefer A to B and B to C, you
[04:40] prefer A to C.
[04:41] Okay?
[04:42] Uh
[04:44] that's that's kind of um
[04:46] uh I'm sure that's pretty clear. You've
[04:47] done this a lot in other classes.
[04:49] So, these two are sort of standard
[04:51] assumptions you might make in any math
[04:53] class.
[04:54] The third assumption is the one where
[04:55] the economics comes in,
[04:57] which is the assumption of
[04:58] non-satiation,
[05:02] or the assumption of more is better.
[05:08] In this class, we will assume more is
[05:12] always better than less.
[05:15] Okay? We'll assume more is better than
[05:16] less. Now, to be clear, we're not going
[05:19] to say that the next unit makes you
[05:21] equally happy as the last unit. In fact,
[05:23] I'll talk about that in a few minutes.
[05:24] We'll in fact assume it makes you less
[05:26] happy.
[05:27] But, we will say you always want more.
[05:29] The face of the chance of more or less,
[05:31] you'll always be happier with more.
[05:33] Okay? And that's the non-satiation
[05:35] assumption.
[05:36] Okay? I'll talk about that some during
[05:38] the lecture, but that's sort of what's
[05:40] going to give our models their power.
[05:41] That's the sort of new economics
[05:43] assumption that's going to give, beyond
[05:45] our typical math assumptions, that's
[05:46] going to give our models their power.
[05:48] Okay?
[05:49] So, that's our assumptions.
[05:51] So, armed with those,
[05:54] I want to start with the graphical
[05:55] representation of preferences.
[05:58] I want to graphically represent people's
[05:59] preferences. And I'll do so through
[06:00] something we call indifference curves.
[06:07] Indifference curves.
[06:10] Okay?
[06:11] These are base Indifference curves are
[06:12] basically preference maps.
[06:15] Essentially, indifference curves are
[06:16] graphical maps of preferences.
[06:19] Okay?
[06:20] So, for example,
[06:23] suppose your parents gave you some money
[06:26] at the beginning of the semester, and
[06:27] you can spend that money on two things.
[06:29] Your parents are rich. They give you
[06:29] tons of money. Spend that money on two
[06:32] things.
[06:33] Buying pizza
[06:34] or or um or eating cookies.
[06:38] Okay?
[06:39] So, consider your Consider preferences
[06:42] between pizza and cookies. That's your
[06:43] two things you can do. Once again, it's
[06:45] constrained model. Obviously, in life
[06:46] you can do a million things with your
[06:47] money. But, it turns out if we consider
[06:50] the contrast between doing two different
[06:52] things with your money, you get a rich
[06:53] set of intuition that you can apply to a
[06:55] much more multi-dimensional decision
[06:57] case.
[06:57] So, let's start with the two-dimensional
[06:59] decision case. You've got your money.
[07:00] You're going to have pizza or you're
[07:02] going to have cookies.
[07:03] Okay? Now, consider three choices. Okay?
[07:06] Choice A
[07:09] is two pizzas
[07:10] and one cookie.
[07:13] Choice B
[07:15] is one pizza,
[07:18] one pizza and two cookies, and choice C
[07:22] is two pizzas, two cookies.
[07:25] Okay? That's the three packages I want
[07:26] to compare.
[07:28] And I'm going to assume and I'll
[07:30] mathematically rationalize in a few
[07:32] minutes. But for now, I'm going to
[07:33] assume you are indifferent
[07:37] between these two packages.
[07:39] I'm going to assume you're equally happy
[07:40] with two slices of pizza and one cookie
[07:42] or two cookies and one slice of pizza.
[07:45] Okay?
[07:46] I'm going to assume that. But, I'm also
[07:49] going to assume you prefer option C to
[07:51] both of these.
[07:53] In fact, I'm not I'm going to assume
[07:54] that because that is what more is better
[07:56] gives you.
[07:57] Okay? So, you're indifferent between
[07:59] these.
[08:00] This indifference doesn't come from any
[08:01] property I wrote up there. That's an
[08:02] assumption. That's just a I just for
[08:04] this case I'm assuming that. This comes
[08:06] from the third property I wrote up
[08:07] there. You prefer package C cuz more is
[08:09] always better than less.
[08:11] Okay?
[08:12] So, now let's graph your preferences.
[08:15] And we do so in figure 2.1
[08:18] Okay? In the handout. Um
[08:20] Okay. So, um
[08:23] here's your indifference curve. So,
[08:24] we've graphed on the x-axis your number
[08:26] of number of cookies. On the y-axis,
[08:28] slices of pizza.
[08:30] Okay? Now, you have We've graphed the
[08:33] three choices I laid here. Choice A,
[08:36] which is two slices of pizza and one
[08:37] cookie.
[08:39] Choice B, which is two cookies and one
[08:40] slice of pizza. And choice C, which is
[08:42] two of both.
[08:44] And I have drawn on this graph your
[08:45] indifference curves. The way your
[08:47] indifference curves looks is there's one
[08:49] indifference curve between A and B
[08:51] because those are the points among which
[08:53] you're indifferent.
[08:54] So, what an indifference curve
[08:56] represents is all combinations of
[08:59] consumption among which you are
[09:01] indifferent. So, we call it indifference
[09:03] curve.
[09:03] So, an indifference curve, which will be
[09:05] sort of one of the big workhorses of
[09:07] this course. An indifference curve
[09:09] represents all combinations along which
[09:12] you are indifferent. You are indifferent
[09:13] between A and B. Therefore, they lie on
[09:15] the same curve.
[09:17] Okay?
[09:19] So, that's sort of our our our
[09:21] preference map, our indifference curves.
[09:23] And these indifference curves are going
[09:24] to have four properties.
[09:27] Four properties that you have to that
[09:30] follow naturally from this It's really
[09:32] three and a half. The third and fourth
[09:33] are really pretty much the same, but I
[09:35] like to write them out as four.
[09:36] Four properties that follow from the
[09:38] from these underlying assumptions.
[09:41] Property one
[09:42] is consumers prefer higher indifference
[09:45] curves.
[09:46] Consumers prefer
[09:49] higher
[09:51] indifference curves.
[09:53] Okay? And that just follow from more is
[09:55] better. That is, an indifference curve
[09:56] that's higher goes through the package
[09:58] that has at least as much of one thing
[10:00] and more of the other thing. Therefore,
[10:02] you prefer it.
[10:03] Okay? So, as indifference curves shift
[10:05] out, people are happier.
[10:08] Okay? So, on that higher indifference
[10:09] curve
[10:11] point C, you are happier than points A
[10:13] and B because more is better.
[10:15] Okay?
[10:17] The second
[10:19] is that
[10:21] indifference curves
[10:23] never cross.
[10:27] Indifference curves
[10:29] uh never cross.
[10:31] Okay, actually that's
[10:33] I'm going to that's third actually. I
[10:35] want to come to that in order. Second,
[10:37] third is indifference curves never
[10:38] cross. Second is indifference curves are
[10:39] downward sloping.
[10:44] Second is indifference curves are
[10:45] downward sloping.
[10:47] Okay? Indifference curves are downward
[10:48] sloping. Let's talk about that first.
[10:51] Okay?
[10:52] That simply comes from the principle of
[10:54] non-satiation.
[10:57] So, look at figure 2.2.
[10:59] Here's an upward sloping indifference
[11:00] curve.
[11:03] Okay? Why does that violate the
[11:05] principle of non-satiation? Why does
[11:06] that violate that? Yeah.
[11:07] Well, either if you're either you're
[11:09] somehow less happy if you have more
[11:11] cookies, like or you're less happy if
[11:13] you have more pizza. Yeah, but
[11:16] And like there's
[11:18] and that violates non-satiation.
[11:20] Exactly. So, basically you're
[11:22] indifferent on this curve you're
[11:23] indifferent between one of each and two
[11:24] of each. You can't be indifferent. Two
[11:26] of each has got to be better than one of
[11:27] each.
[11:28] So, an upward sloping indifference curve
[11:29] would violate non-satiation.
[11:31] So, that's the second property of
[11:32] indifference curve.
[11:34] The third property of indifference curve
[11:35] is indifference curves never cross.
[11:37] Okay? We can see that
[11:40] in figure 2.3.
[11:42] Okay? Someone else tell me why this
[11:44] violates the properties I wrote up there
[11:45] indifference curves crossing.
[11:48] Yeah.
[11:49] Because B and C
[11:51] What's that? Because B and C are
[11:53] strictly better.
[11:54] Because the B and C, B is strictly
[11:56] better. That's right.
[11:58] But but they're they're also but they're
[12:00] also both on the same curve as A. So,
[12:03] you're saying they're both you're
[12:04] indifferent with A for both B and C, but
[12:06] you can't be because B is strictly
[12:07] better than C. So, it violates
[12:09] transitivity.
[12:10] Okay? So, the problem with crossing
[12:12] indifference curves they violate
[12:13] transitivity.
[12:17] And then finally, the fourth
[12:20] is sort of a cute extra assumption, but
[12:22] I think it's important to clarify,
[12:24] which is that there is only one
[12:27] indifference curve through every
[12:29] possible consumption bundle.
[12:33] Only one
[12:34] IC
[12:36] through
[12:37] every bundle.
[12:42] Okay? You can't have two indifference
[12:43] curves going through the same bundle.
[12:45] Okay? Uh and that's because of
[12:47] completeness. If you have two
[12:49] indifference curves going through the
[12:50] same bundle, you wouldn't know how you
[12:51] felt.
[12:52] Okay? So, there can only be one going
[12:54] through every bundle cuz you know how
[12:55] you feel.
[12:56] You may feel indifferent, but you know
[12:57] how you feel. You can't say I don't
[12:59] know.
[13:00] Okay? So, that's sort of a extra
[13:01] assumption that sort of completes the
[13:03] link to the properties.
[13:05] Okay?
[13:06] So, that's basically how indifference
[13:08] curves work.
[13:10] Now,
[13:11] I find when I I took this course
[13:14] before you were
[13:15] God, maybe before your parents were
[13:16] born. I don't know. Certainly before you
[13:18] guys were born. Okay, when I took this
[13:19] course,
[13:20] I found this course full of a lot of
[13:22] lightbulb moments. That is stuff was
[13:23] just sort of confusing then boom an
[13:24] example really make it work for me. And
[13:26] the example that made indifference
[13:27] curves work to me was actually during my
[13:29] first year up.
[13:31] When my year up was with a grad student
[13:33] and that grad student had to decide
[13:34] where he was going to accept a job. He
[13:35] had a series of job offers and he had to
[13:36] decide. And basically said, here's the
[13:38] way I'm thinking about it. I'm
[13:40] indifferent I've I've indifference map
[13:42] and I care about two things. I care
[13:44] about school location
[13:47] and I care about economics department
[13:49] quality.
[13:53] Okay? I care about the quality of my
[13:54] colleagues and the research that's done
[13:55] there and the location. And basically
[13:57] he's he had two offers.
[13:59] One was from Princeton, which he put up
[14:02] here. No offense to New Jerseyans, but
[14:03] Princeton as a young single person
[14:04] sucks. Okay, fine when you're married
[14:06] and have kids, but deadly as a young
[14:08] single person. And the other so, that's
[14:10] Princeton. Down here was Santa Cruz.
[14:13] Okay? Awesome can have the most
[14:15] beautiful university in America. Okay?
[14:17] But not as good an economics department.
[14:19] And he decided he was roughly
[14:20] indifferent between the two.
[14:23] But he had a third offer
[14:25] from the IMF, which is a research
[14:27] institution in DC, which as he had a lot
[14:29] of good colleagues and DC is way better
[14:31] than Princeton, New Jersey even if it's
[14:32] not as good as Santa Cruz. So, he
[14:34] decided he would take the offer at the
[14:35] IMF.
[14:37] Okay? Even though the IMF had worse
[14:39] colleagues than Princeton
[14:41] and worse location than Santa Cruz,
[14:43] it was still better in combination the
[14:45] two of them given his preferences.
[14:47] And that's how he used indifference
[14:48] curves to help make his decision.
[14:50] Okay?
[14:52] So, that's sort of an example of uh of
[14:54] applying it. Let's see no offense to the
[14:55] New Jerseyans in the room, of which I'm
[14:57] one, but believe me you'd rather be in
[14:59] Santa Cruz.
[15:00] Um okay. Uh
[15:02] so, now let's go from preferences
[15:05] to utility functions.
[15:12] Okay? So, now we're going to move from
[15:13] preferences,
[15:15] which we've represented graphically,
[15:17] to utility functions, which we're going
[15:19] to represent mathematically. Remember, I
[15:21] want you to understand everything in
[15:22] this course at three levels:
[15:23] graphically, mathematically, and most
[15:25] importantly of all intuitively. Okay?
[15:28] So, graphic is indifference curves. Now
[15:30] we'll come to the mathematical
[15:31] representation, which utility function.
[15:34] Okay? And the idea is that every
[15:36] individual, all of you in this room,
[15:38] have a stable, well-behaved underlying
[15:41] mathematical representation of your
[15:43] preferences,
[15:44] which we call utility function.
[15:47] Okay? Now, once again, that's going to
[15:49] be very complicated and you have
[15:50] preference over lots of different
[15:51] things. We're going to make things
[15:52] simple by writing out two-dimensional
[15:54] representation for now of your
[15:56] indifference curve. We're going to say,
[15:58] how do we mathematically represent your
[16:00] feelings about pizza versus cookies?
[16:03] Okay? Imagine that's all you care about
[16:05] in the world is pizza and cookies.
[16:07] How do we mathematically represent that?
[16:09] So, for example, we could write down
[16:11] that your utility function is equal to
[16:14] the square root of the number of slices
[16:15] of pizza times the number of cookies.
[16:18] We could write that down. I'm not saying
[16:19] that's right. I'm not saying it works
[16:21] for anyone in this room or even everyone
[16:22] in this room, but that is a possible way
[16:25] to represent utility.
[16:28] Okay? What this would say, this is
[16:30] convenient. We will use we'll end up
[16:32] using square root form a lot for utility
[16:33] functions, a lot of convenient
[16:34] mathematical properties.
[16:36] And it happens to jive with our example,
[16:39] right? Because in this example you're
[16:40] indifferent between two pizza and one
[16:42] cookie or one pizza and two cookie.
[16:44] They're both square root of two.
[16:45] And you prefer two pizza and two
[16:47] cookies, that's two.
[16:48] Okay? So, this gives you a higher
[16:50] utility for two pizza and two cookies
[16:53] okay? Um uh
[16:55] then uh one pizza
[16:57] than one pizza and two cookie or two
[16:59] pizza and one cookie.
[17:02] So, now the question is what does this
[17:03] mean? What is utility?
[17:05] Okay? Well, utility doesn't actually
[17:08] mean anything. There's not really a
[17:10] thing out there called utils.
[17:12] Okay?
[17:14] In other words, utility is not a
[17:15] cardinal concept. It is only an ordinal
[17:18] concept.
[17:19] You cannot say your utility you are you
[17:23] cannot literally say my utility is X%
[17:26] higher than your utility, but you can
[17:27] rank them.
[17:29] So, we're going to assume that utility
[17:31] can be ranked to allow you to rank
[17:33] choices.
[17:34] Even if generally we might slip some and
[17:36] sort of pretend utility is cardinal for
[17:38] some cute examples, but by and large
[17:41] we're going to think of utility as
[17:42] purely ordinal. It's just a way to rank
[17:44] your choices. It's just when you have a
[17:46] set of choices out there over many
[17:47] dimensions. Like if your choice in life
[17:49] is always over one dimension
[17:51] and more was better, it would always be
[17:53] easy to rank it, right? You never ever
[17:54] problem. Once your choice over more than
[17:56] one dimension,
[17:58] now if you want to rank them, you need
[18:00] some way to combine them. That's what
[18:01] this function does. It allows you
[18:03] essentially to weight the different
[18:05] elements of your consumption bundle so
[18:07] you can rank them um
[18:09] when it comes time to choose.
[18:11] Okay? Now, this is obviously incredibly
[18:14] simple,
[18:15] but it turns out to be amazingly
[18:17] powerful in explaining real world
[18:19] behavior.
[18:20] Okay?
[18:21] And so, what I want to do today is work
[18:23] with the underlying mathematics of
[18:24] utility
[18:25] and then we'll come back we'll see in
[18:26] the next few lectures how it could
[18:28] actually be used to explain decisions.
[18:31] So, a key concept we're going to talk
[18:33] about
[18:34] in this class is marginal utility.
[18:38] Marginal utility is just the derivative
[18:39] of the utility function with respect to
[18:41] one of the elements. So, the marginal
[18:43] utility for cookies of cookies is
[18:45] utility of the next cookie
[18:48] given how many cookies you've had.
[18:50] This class can be very focused on
[18:52] marginal decision making.
[18:55] In economics, it's all about how you
[18:56] think about the next unit. Turns out
[18:58] that makes life a ton easier. Turns out
[19:00] it's way easier to say, do you want the
[19:02] next cookie? Than to say, how many
[19:03] cookies do you want?
[19:06] Because if you want the next cookie,
[19:07] that's sort of a very isolated decision.
[19:09] You say, okay, I've had this many
[19:10] cookies, do I want the next cookie?
[19:12] Whereas before you start eating you say,
[19:13] how many cookies do you want? That's
[19:14] sort of a harder more global decision.
[19:16] So, we're going to focus on the stepwise
[19:18] decision making process of do you want
[19:20] the next unit, the next cookie or the
[19:22] next slice of pizza?
[19:24] Okay? And the key feature of utility
[19:28] functions we'll work with throughout the
[19:29] semester
[19:31] is that they will feature diminishing
[19:36] diminishing marginal utility.
[19:39] Marginal utility will fall as you have
[19:41] more of a good.
[19:43] The more of a good you've had, the less
[19:45] happiness you'll derive from the next
[19:47] unit.
[19:48] Okay?
[19:50] Now, we can see that graphically in
[19:52] figure 2-4.
[19:55] Figure 2-4 graphs on the x-axis the
[19:58] number of cookies holding constant
[20:00] pizza. So, let's say you're having two
[20:01] pizza slices and you want to say,
[20:04] "What's my benefit from the next
[20:05] cookie?"
[20:07] And on the on the left axis, violating
[20:09] what I just said like 15 seconds ago, we
[20:11] graph utility.
[20:13] Now, once again, the util numbers don't
[20:14] mean anything. It's just to give you an
[20:16] ordinal sense.
[20:18] What you see here is that if you have
[20:19] one cookie,
[20:20] your utility is 1.4, square root of 2 *
[20:23] 1.
[20:25] If you have two cookies, your utility
[20:26] goes up
[20:28] to square root of 4, which is 2. You are
[20:30] happier with two cookies.
[20:32] But you are less happy from the second
[20:34] cookie than the first cookie.
[20:36] Okay? And you can see that in figure if
[20:39] you flip back and forth between 2-4 and
[20:40] 2-5, you can see that.
[20:43] Okay? The first cookie,
[20:46] going from zero to one cookie, gave you
[20:48] one So, in this case, we're not graphing
[20:50] the marginal utility. So, figure 2-4 is
[20:53] the level of utility,
[20:56] which is not really something you can
[20:57] measure, in fact. Figure 2-5 is
[20:59] something you can measure, which is
[21:00] marginal utility. What's your happiness?
[21:02] And we'll talk about measuring this from
[21:03] the next cookie. You see, the first
[21:05] cookie
[21:06] gives you
[21:08] um a utility and a utility increment of
[21:10] 1.4.
[21:13] Okay? You go from utility of zero
[21:15] to utility of 1.4.
[21:17] The next cookie gives you a utility
[21:18] increment of 0.59.
[21:21] Okay, you go from utility of 1.41
[21:23] to utility of 2.
[21:25] The next cookie gives you a utility
[21:26] increment of 0.45, the square root of 3.
[21:28] So, now we flip back to the previous
[21:30] page. We're going from the square root
[21:32] of 4 to We're going from the square root
[21:34] of 4, I'm sorry, to the square root of
[21:35] 6.
[21:36] Square root of 6 is only 0.45 more than
[21:39] the square root of 4.
[21:40] And so on.
[21:42] So, each additional cookie
[21:44] makes you less and less happy. It makes
[21:46] you happier. It has to cuz more is
[21:48] better. But it makes you less and less
[21:50] happy.
[21:51] Okay? And this makes sense.
[21:54] Just think about any decision in life
[21:56] starting with nothing of something and
[21:58] having the first one. Slice of pizza, a
[22:00] cookie, deciding on which movie to go
[22:02] to.
[22:03] The first movie, the one you want to see
[22:04] the most,
[22:06] okay, is going to make you happier than
[22:08] you want to see the one you want to see
[22:09] not quite as much.
[22:11] The first cookie when you're hungry will
[22:12] make you happier than the second cookie.
[22:15] The first slice of pizza will make you
[22:16] happier. Now, you may be close to
[22:17] indifferent where that second slice of
[22:19] pizza makes you almost as happy as the
[22:20] first.
[22:21] But the first will make you happier.
[22:23] Okay, if you think about that's really
[22:25] sort of that first step. You were hungry
[22:27] and that first one makes you feel
[22:27] happier.
[22:29] Now, but you got to remember you always
[22:32] want more cookies.
[22:33] Now, you might say, "Wait a second, this
[22:35] is stupid. Okay, once I've had 10
[22:36] cookies, I'm going to barf."
[22:38] The 11th cookie could actually make me
[22:39] worse off cuz I don't like barfing.
[22:42] But
[22:43] in economics, we have to remember you
[22:45] don't have to eat the 11th cookie. You
[22:47] could give it away.
[22:49] So, if I say you want to buy the 11th
[22:50] cookie, you could save it for later. You
[22:52] could give it to a friend.
[22:53] So, you always want it. In the worst
[22:56] case, you throw it out.
[22:57] It can't make you worse off.
[22:59] It can only make you better off.
[23:01] And that's what where our sort of more
[23:03] is better assumption comes from.
[23:04] Obviously, in the limit, you know, if
[23:05] you get a million cookies, your garbage
[23:07] can gets full, you have no friends to
[23:08] give them to. I understand the limit
[23:09] these things fall apart. Okay? But
[23:12] that's the basic idea of more is better
[23:13] and the basic idea of of diminishing
[23:15] marginal utility. Okay, any questions
[23:16] about that?
[23:17] Yeah.
[23:20] Utility function can never be negative
[23:22] because we have Well, utility Once
[23:25] again, utility is not an ordinal
[23:26] concept. You can set up utility
[23:28] functions such that the number is
[23:29] negative. You can set that up. Okay? The
[23:31] marginal utility is always positive. You
[23:34] always get some benefit from the next
[23:35] unit.
[23:37] Utility, once again, the measurement is
[23:38] irrelevant. So, it can be negative. You
[23:39] can set it up. Yeah, I could write my
[23:40] utility function like this, you know,
[23:42] something like that. So, it could be
[23:43] negative. That's just a sort of scaling
[23:45] factor. But marginal utility is always
[23:47] positive. You're always happier or it's
[23:49] not it's not negative. You're always
[23:51] happier at least indifferent to getting
[23:53] the next unit.
[23:54] Yeah. So, when you're looking at 2-5,
[23:56] you can't look at the function of the
[23:58] pizza and the marginal utility, so it's
[23:59] going to go down.
[24:02] Uh I'm sorry. You look Figure 2-5, no,
[24:04] the marginal utility is going to go
[24:05] down.
[24:06] Each fraction of cookie, the marginal
[24:07] utility Marginal utility is always
[24:09] diminishing. So, if you start with zero,
[24:11] then you can get half of the pizza in
[24:12] this graph.
[24:14] Well, it's really hard to do it from
[24:15] zero. That's really tricky. It's sort of
[24:17] much easier to start from one.
[24:19] So, corner solutions, we'll talk a lot
[24:20] about corner solutions in this class.
[24:21] They get ugly. Think of it starting from
[24:23] one. Starting from that first cookie,
[24:25] every fraction of a cookie makes you
[24:26] happier but less and less happy with
[24:27] each fraction. It's a good question.
[24:29] All right. Good questions. All right.
[24:31] So, now let's let's talk about Let's
[24:35] flip back from the math to the graphics
[24:38] and talk about where indifference curves
[24:40] come from. I just drew them out. But in
[24:42] fact, indifference curves are the
[24:44] graphical representation of what comes
[24:46] out of utility function.
[24:49] Okay? And indeed, the slope of the
[24:51] indifference curve we are going to call
[24:54] the marginal rate of substitution.
[24:58] The rate essentially at which you're
[24:59] willing to substitute
[25:02] one good for the other. The rate at
[25:04] which you're willing to substitute
[25:05] cookies for pizza is your marginal rate
[25:09] of substitution.
[25:11] And we'll define that as the slope of
[25:14] the indifference curve, delta P over
[25:16] delta C.
[25:18] That is your marginal rate of
[25:19] substitution. Literally, the
[25:20] indifference curve tells you the rate at
[25:22] which you're willing to substitute. You
[25:23] just follow along and say, "Look, I'm
[25:25] willing to give up um So, in other
[25:28] words, if you look at figure 2-6,
[25:30] you say, "Look, I'm indifferent between
[25:33] point A and point B.
[25:35] One cook- one slice of pizza and I'm
[25:37] sorry, one cookie and four slices of
[25:39] pizza
[25:40] is the same to me as two cookies and two
[25:42] slices of pizza. Why is it the same?
[25:44] Because they both give me utility square
[25:45] root of 4, right? So, given this
[25:47] mathematical rep- I'm not saying you
[25:49] are. I'm saying given this mathematical
[25:50] representation,
[25:52] okay, you are indifferent between point
[25:54] A and point B.
[25:55] So, what that says and what's the slope
[25:57] of the indifference curve? What's the
[25:58] arc slope between point A and point B?
[26:01] The slope is -2.
[26:04] So, your marginal rate of substitution
[26:05] is -2.
[26:07] You are indifferent.
[26:09] Okay?
[26:10] Um you're indifferent between 1-4 and
[26:13] 2-2. Therefore, you're willing to
[26:15] substitute or give away
[26:18] two slices of pizza to get one cookie.
[26:22] Delta P delta C
[26:24] is uh
[26:25] is -2.
[26:28] Okay? Now, it turns out you can define
[26:31] the marginal rate of substitution over
[26:32] any segment of indifference curve. And
[26:34] what's interesting is it changes. It
[26:36] diminishes. Look what happens when we
[26:39] move from two pizzas and two cookies
[26:42] to from point B to point C.
[26:44] Now, the marginal rate of substitution
[26:46] is only -1/2.
[26:47] Now, I'm only willing to give up one
[26:50] slice of pizza to get two cookies.
[26:52] What's happening? First, I was willing
[26:53] to give up two slices of pizza to get
[26:55] one cookie.
[26:57] Now, I'm only willing to give up willing
[26:58] to give up one slice of pizza to get two
[27:00] cookies. What's happening?
[27:02] Yeah. You don't want a cookie as much.
[27:03] Because of? Diminishing marginal
[27:05] utility. Exactly. Diminishing marginal
[27:07] utility has caused the marginal rate of
[27:10] substitution itself itself to diminish.
[27:12] For those of you who are really kind of
[27:13] better at math than I am, it turns out
[27:15] technically, mathematically, marginal
[27:17] utility isn't always diminishing. You
[27:19] can draw cases. MRS is always
[27:22] diminishing.
[27:23] So, you can think of marginal utility as
[27:24] always diminishing. It's fine for this
[27:25] class. When you get to higher level math
[27:27] and economics, you'll see marginal
[27:28] utility doesn't have to diminish. MRS
[27:30] has to diminish.
[27:32] Okay? MRS is always diminishing.
[27:35] As you go along the indifference curve,
[27:36] that slope is always falling.
[27:40] Okay?
[27:41] So, basically,
[27:43] what we can write now is how the MRS
[27:46] relates to utility function,
[27:48] our first sort of mind-blowing result,
[27:50] is that the MRS is equal to the negative
[27:53] of the marginal utility of cookies over
[27:56] the marginal utility of pizza.
[27:59] That's our first key definition.
[28:02] It's equal to the negative of the
[28:04] marginal utility of the good on the
[28:05] x-axis over the marginal utility of the
[28:07] good on the y-axis.
[28:10] Okay? Essentially, the marginal rate of
[28:13] substitution tells you how your relative
[28:16] marginal utilities evolve as you move
[28:19] down the indifference curve.
[28:22] When you start at point A,
[28:25] you have lots of pizza and not a lot of
[28:28] cookies.
[28:30] When you have lots of pizza,
[28:33] your marginal utility is small.
[28:36] Here's the key insight. This is the
[28:38] thing which, once again, it's a light
[28:40] bulb thing. If you get this, it'll make
[28:41] your life so much easier. Marginal
[28:43] utilities are negative functions of
[28:46] quantity.
[28:47] The more you have of a thing, the less
[28:50] you want the next unit of it.
[28:53] That's why, for example, cookies is now
[28:54] in the numerator and pizza is in the
[28:55] denominator, flipping from this side.
[28:58] Okay? The more you have a good, the less
[29:00] you want it.
[29:02] So, start at point A. You have lots of
[29:04] pizza and not a lot of cookies.
[29:07] You don't really want more pizza. You
[29:09] want more cookies.
[29:10] That means the denominator is small.
[29:13] The marginal utility of pizza is small.
[29:16] You don't really want it.
[29:18] But the marginal utility of cookies is
[29:19] high. You don't have many of them. So,
[29:21] this is a big number.
[29:24] Now, let's move to point B.
[29:26] And think about your next decision.
[29:28] Well, now
[29:30] your marginal utility of pizza,
[29:33] if you're going to go from two to one
[29:34] slice of pizza, now pizza is worth a lot
[29:35] more than cookies. So, now it gets
[29:37] smaller.
[29:38] So, essentially, as you move along that
[29:40] indifference curve, because of this you
[29:43] want because diminishing marginal
[29:44] utility,
[29:45] it leads this issue of a diminishing
[29:47] marginal rate of substitution.
[29:49] Okay? So, basically, as you move along
[29:52] the indifference curve, you're more and
[29:53] more willing to give up
[29:55] the good on the x-axis to get the good
[29:57] on the y-axis. As you move from the
[29:59] upper left to the lower right on that
[30:00] indifference map, figure 2.6,
[30:03] you're more you're more willing to give
[30:04] up
[30:06] the good on the on the x-axis to get the
[30:08] good on the y-axis.
[30:11] And what this implies is that
[30:13] indifference curves are con-
[30:16] indifference curves
[30:17] are uh convex to the origin.
[30:21] Indifference curves are convex to the
[30:22] origin.
[30:23] It's very important. Okay, the
[30:26] Let's see. The They are They are not
[30:28] concave. They're either convex or
[30:29] straight. Let's say they're They're not
[30:31] concave to the origin.
[30:33] Okay, to be technical.
[30:34] Indifference curves can be linear. We'll
[30:36] come to that.
[30:37] But they can't be concave to the origin.
[30:39] Why? Well, let's look at the next
[30:40] figure, the last figure, figure 2.7.
[30:43] What would happen if indifference curves
[30:44] were concave to the origin?
[30:47] Then that would say moving from
[30:51] one pizza, so now I've drawn a a concave
[30:54] indifference curve. And with this
[30:55] indifference curve, moving from point A
[30:57] to point B leaves you indifferent.
[31:00] So, you're happy to give up one slice of
[31:02] pizza to get one cookie.
[31:04] Starting with four slices of pizza and
[31:06] one cookie,
[31:07] you were happy to give up one slice of
[31:09] pizza to get one cookie.
[31:12] Now, starting from two and three, you're
[31:15] now willing to give up two slices of
[31:16] pizza to get one cookie.
[31:19] What does that violate? Why Why does
[31:20] that not make sense? Yeah.
[31:23] Law of diminishing marginal returns.
[31:25] Yeah, it's law of diminishing marginal
[31:26] utility. Here, you were you were you
[31:28] were only You were happy to have one
[31:30] slice of pizza to get one cookie. Now
[31:31] you were willing to have two slices of
[31:32] pizza to get one cookie, even though you
[31:33] have less pizza and more cookies. That
[31:35] can't be right. As you have less pizza
[31:37] and more cookies, cookies pizza should
[31:40] become more valuable, not less valuable.
[31:42] And cookies should become less valuable,
[31:43] not more valuable.
[31:45] So, a concave to the origin indifference
[31:47] curve would violate the principle of
[31:49] diminishing marginal utility and
[31:50] diminishing marginal rate of
[31:51] substitution.
[31:53] Okay? Yeah. What if it's like trading
[31:55] cards?
[31:56] Okay. I mean, theory and
[31:58] I mean, I think you get more of trading
[32:00] cards,
[32:01] you you you have fewer cards than you
[32:03] want to
[32:04] trade that.
[32:05] That's very interesting. So, in some
[32:06] sense,
[32:08] what that is saying is that your utility
[32:10] function is really over sets. You're
[32:11] saying your utility function is over
[32:12] trading cards. It's over sets.
[32:15] So, basically, that's what sort of you
[32:17] know, what's sort of a bit
[32:19] you know, our models are flexible. One
[32:22] way to say they're loose, another way to
[32:23] say they're flexible. So, one But one of
[32:26] the challenges you'll face on this
[32:27] course is thinking about what is
[32:30] decision set over which I'm writing my
[32:31] utility function. You're saying it's
[32:32] sets, not trading cards. So, that's why
[32:34] it happens.
[32:36] Okay? Other questions? Good question.
[32:38] Yeah, in the back. What about like
[32:39] addictive things where like the more you
[32:41] have of it, the more you want to buy?
[32:43] Yeah, that's that's a really really good
[32:45] question. I spent a lot of my research
[32:46] life, actually. I do a lot of I did a
[32:48] lot of research for a number of years on
[32:50] thinking about how you properly model
[32:52] addictive decisions like smoking.
[32:54] Addictive decisions like smoking,
[32:56] essentially, it really is that your
[32:59] utility function itself shifts as you
[33:02] get more addictive.
[33:03] It's not that your marginal utility, the
[33:06] next cigarette is still worth less than
[33:07] the first cigarette. It's just that as
[33:09] you get more addicted, that first
[33:10] cigarette gets worth more and more to
[33:12] you.
[33:13] So, when you wake up in the morning
[33:14] feeling crappy, that first cigarette
[33:16] still does more for you than the second
[33:17] cigarette. It's just the next day you
[33:19] wake up feeling crappier.
[33:21] Okay? So, we model addiction as
[33:23] something where essentially each day
[33:26] cigarettes do less and less for you. You
[33:28] get essentially adjusted to new You get
[33:29] habituated to higher levels.
[33:31] And this is why, you know, I do a lot of
[33:33] work, you know, this is why,
[33:34] unfortunately, we saw last year the
[33:36] number of the highest number of deaths
[33:37] from accidental overdose in US history,
[33:39] 72,000 people died from drug overdoses
[33:42] last year, more than ever died in
[33:43] traffic accidents in our nation's
[33:44] history.
[33:45] Okay? Why?
[33:47] Because people get habituated to certain
[33:49] levels. And they used to be used to
[33:51] certain levels. So, people get hooked on
[33:52] OxyContin.
[33:54] They get habituated to a certain level.
[33:55] They maybe switch to heroin. And they're
[33:57] habituated to a certain level. And now
[33:58] there's this thing called fentanyl,
[33:59] which is synthetic opioid brought over
[34:01] from China, which is incredibly
[34:03] powerful. And dealers are mixing the
[34:05] fentanyl in with the heroin.
[34:07] And the people shoot up not realizing at
[34:09] their habituated level, not realizing
[34:11] they have this dangerous substance, and
[34:12] they overdose and die.
[34:14] And that's because they've got
[34:15] habituated to higher levels. They didn't
[34:16] realize they're getting a different
[34:17] product. So, it's not about not
[34:18] diminishing marginal utility. It's about
[34:19] different underlying different products.
[34:22] All right?
[34:23] Other questions?
[34:25] Sorry for that depressing note, but it's
[34:27] important to be thinking about That's
[34:28] why, once again, we're the dismal
[34:29] science. We have to think about these
[34:30] things. Okay. Now, let's come to a great
[34:33] example
[34:34] that I hope you've wondered about, and
[34:36] maybe you've already figured out in your
[34:37] life. But I hope you've at least stopped
[34:39] and wondered about,
[34:41] which is the prices of different sizes
[34:44] of goods
[34:45] in a convenience store, say.
[34:48] Okay?
[34:49] Take Starbucks.
[34:51] You can get a tall iced coffee for
[34:53] $2.25.
[34:55] Or the next size, whatever the hell they
[34:56] call it, bigger. Okay? You can get for
[35:00] 70 more cents. So, $2.25 and you can
[35:03] double it for 70 more cents. Or take
[35:05] McDonald's. A small drink is $1.22
[35:09] at the local McDonald's. But for 50 more
[35:11] cents, you can double the size.
[35:14] Okay? What's going on here?
[35:16] It Why do they give you twice as much
[35:18] liquid?
[35:19] Or if you go for ice cream, it's the
[35:20] same thing. Why do they give you twice
[35:22] as much for much less than twice as much
[35:25] money?
[35:26] What's going on? Yeah.
[35:27] Um since your marginal utility is is
[35:30] diminishing as you have more coffee
[35:33] available to you, you're willing to pay
[35:34] less for it. So, they make like the
[35:37] additional coffee cheaper.
[35:39] Exactly. That's a great way to explain
[35:41] it. The point is it's all about
[35:43] diminishing marginal utility.
[35:45] Okay? When you come in to McDonald's on
[35:47] a hot day, you are desperate for that
[35:48] soda.
[35:50] But you're not as desperate to have
[35:51] twice as much soda. You'd like it.
[35:52] You're probably willing to pay more for
[35:54] it. But you don't like it nearly as much
[35:56] as that first bit of soda.
[35:58] So, those prices simply reflect the
[36:00] market's reaction to understanding
[36:02] diminishing marginal utility.
[36:04] Now, we haven't talked about the supply
[36:06] side of the market yet. I'm not getting
[36:07] to how providers make decisions. That's
[36:09] a much deeper issue. I'm just saying
[36:11] that this is diminishing marginal
[36:12] utility in action, how it works in the
[36:14] market. And that's why you see this.
[36:17] Okay? So, basically, um what you see is
[36:21] that uh that first bite of ice cream,
[36:23] for example, is worth more, and that's
[36:25] why the ice cream is twice as big
[36:27] doesn't cost uh twice as much.
[36:30] Now, so basically, what this means is if
[36:34] you think about our demand and supply
[36:36] model,
[36:37] on a hot day, or any day,
[36:40] the demand for the first 16 oz
[36:43] is higher than the demand for the second
[36:46] 16 oz.
[36:48] But the cost of producing 16 oz is the
[36:50] same. So, let's think about this. It's
[36:52] always risky when I try to draw a graph
[36:54] on the board, but let's bear with me.
[36:56] Okay? So, let's say we have this sort of
[36:57] simple supply and demand model.
[36:59] You have this You have this supply
[37:01] function for soda. And let's assume it's
[37:04] roughly flat. Okay, let's assume sort of
[37:06] the cost of firm producing each, you
[37:08] know, within some range, the firm
[37:10] basically every incremental 16 oz costs
[37:13] them the same. So, that's sort of their
[37:14] supply curve. Okay? And then you have
[37:17] some demand curve.
[37:18] Okay? You have some demand curve, which
[37:20] is downward sloping. Okay? And they set
[37:23] some price. And this is the demand for
[37:25] 16 oz.
[37:27] Now, you have What's the demand for the
[37:29] next 16 oz?
[37:31] Okay?
[37:32] Yeah, this isn't going to work.
[37:34] We have to have an upward sloping supply
[37:35] curve. Sorry about that. We have a
[37:36] slightly upward sloping supply curve.
[37:38] Okay? Now we have the demand for the
[37:40] next So, So, here's your Here's your
[37:42] price. Here's your $1.22.
[37:46] Okay? Now you say, "Well, what's my
[37:49] demand when I sell 32 oz?"
[37:52] Well, it turns out demand doesn't shift
[37:53] out twice as much. It just shifts out a
[37:55] little bit more. So, you can only charge
[37:56] $1.72 for the next 16 oz. Probably, if
[38:00] you want to go to the big If you go to
[38:01] the 7-Eleven where you can get sizes up
[38:03] to, you know, as big as your house,
[38:05] okay? They keep These curves keep
[38:07] getting closer and closer to each other.
[38:09] So, those price increments get smaller
[38:11] and smaller. And that's why you get the
[38:12] monster, you know, ginormous gulp at
[38:14] 7-Eleven
[38:16] is really just not that not that
[38:18] different from the price of getting the
[38:20] small little mini size.
[38:21] Okay? Because of diminishing marginal
[38:23] utility.
[38:24] All right? And so, that's how the market
[38:27] That's essentially how we can take this
[38:29] abstract concept, this sort of crazy
[38:32] math, and turn it into literally what
[38:34] you see in the store you walk into.
[38:36] Okay? Questions about that?
[38:38] Yeah.
[38:39] Um so, how does
[38:41] this play into buying in bulk versus
[38:44] buying like a single item?
[38:45] Like if, for example, like you wanted to
[38:48] buy a snack, but you were going to have
[38:49] the breakfast every day. Awesome.
[38:51] Awesome question. And then every single
[38:52] day it was going to be your first
[38:54] granola bar, right? So, so it I I think
[38:57] that its utility would be diminished
[39:00] like every single time. But it's still
[39:02] cheaper to buy in bulk than it would be
[39:04] to buy a single granola bar every Great
[39:06] great question. Yeah. I think that has
[39:08] more to do with packaging costs than
[39:10] with utility.
[39:11] Well, I mean,
[39:13] the risk of my going to this model is,
[39:15] you know, once we once we get non-linear
[39:16] in the world we do things in this class,
[39:17] we have to start talking about supply
[39:18] factors I want to talk to. But there's
[39:20] two answers. One is packaging
[39:21] efficiencies. But the other is if you
[39:24] actually go to Costco
[39:27] and look at their prices, for many
[39:28] things they're not actually better than
[39:29] the supermarket.
[39:31] So, actually, the price of buying the
[39:33] giant like 8,000 bars of granola
[39:36] is actually not that much more
[39:40] than not that much less than a thousand
[39:41] time buying eight pack of eight granola
[39:43] bars.
[39:44] It turns out it's less.
[39:46] But it's not nearly as much less as
[39:47] these examples as sodas and McDonald's.
[39:49] Which is exactly your point. Utility
[39:51] diminishes less.
[39:53] So, they don't want to charge as much
[39:55] less for multiple packages.
[39:57] So, you can actually if you compare
[39:59] perish the gap in perishable product
[40:01] pricing by size, it's much larger than
[40:03] the gap in non-perishable pricing by
[40:05] size. Great point. Yeah. Is there also
[40:07] just like a different time frame for
[40:09] which the utilities start diminishing
[40:11] for every product? Cuz like you gave the
[40:13] example of soda, but it's like would
[40:15] that reset like later in the day if you
[40:17] wanted like were thirsty then? Or
[40:19] Awesome. And that is why they don't let
[40:20] you walk back in with the same cup and
[40:22] refill it.
[40:23] Right? That's exactly right. And that
[40:25] comes this point. It's sort of like it's
[40:27] non-perishable as you get longer apart.
[40:29] Uh so, um but you know, it's all this
[40:32] really interesting thing. So, at Fenway,
[40:34] okay? You can get You get like a regular
[40:37] size soda. It's like crazy. It's like
[40:39] six bucks.
[40:40] Then for like eight bucks, you get a big
[40:42] soda. Then for 10 bucks, you get a
[40:44] refillable big soda. Okay? Now, the
[40:48] question is can you bring that
[40:49] refillable soda back to additional
[40:50] games?
[40:51] Technically not, but I do.
[40:54] Uh and um and basically they sort of
[40:57] understand. So, so there's an
[40:59] interesting question of sort of the
[41:00] perishability of things and how that's
[41:02] and how that's going to affect uh things
[41:04] going on. It's a really it's it's a
[41:05] really it's an interesting question.
[41:07] Other comments?
[41:08] Okay, I'm going to stop there. Those are
[41:10] great comments. Thanks everyone for
[41:11] participating and we'll come back next
[41:13] time and talk about the sad reality that
[41:14] we haven't won the lottery and we have
[41:16] limited amounts of money.